ThesisPDF Available

Strong Primeness in Fuzzy Environment

December 2015

DOI:10.13140/RG.2.2.28783.48808

Thesis for: Doctor of Philosophy Program in Computer Science
Advisor: Regivan Hugo Nunes Santiago

Authors:

Flaulles Boone Bergamaschi

Southwest Bahia State University

Regivan H N Santiago

Federal University of Rio Grande do Norte

The main aim of this investigation is to propose the notion of uniform and strong prime- ness in fuzzy environment. First, it is proposed and investigated the concept of fuzzy strongly prime and fuzzy uniformly strongly prime ideal. As an additional tool, the concept of t/m systems for fuzzy environment gives an alternative way to deal with primeness in fuzzy. Second, a fuzzy version of correspondence theorem and the radical of a fuzzy ideal are proposed. Finally, it is proposed a new concept of prime ideal for Quantales which enable us to deal with primeness in a noncommutative setting.

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Federal University of Rio Grande do Norte

Brazil

Doctoral Thesis

Strong Primeness in Fuzzy Environment

Author:

Flaulles Boone Bergamaschi

Supervisor:

Dr. Regivan H. N. Santiago

A thesis submitted in partial fulﬁllment of the requirements

for the degree of Doctor of Philosophy Program in Computer Science of the

Federal University of Rio Grande do Norte, 2015.

Department of Informatics and Applied Mathematics - DIMAp - UFRN

Doctoral Committee:

Professor Regivan Hugo Nunes Santiago - UFRN, Chair

Professor Benjamín René Callejas Bedregal - UFRN

Professor Elaine Gouvêa Pimentel - UFRN

Professor Edward Hermann Haeusler - PUC-RIO

Professor Laércio Carvalho de Barros - UNICAMP

Bergamaschi, Flaulles Boone.

Strong primeness in fuzzy environment / Flaulles Boone

Bergamaschi. - Natal, 2015.

vii, 65f: il.

Supervisor: Prof. Dr. Regivan Hugo Nunes Santiago.

Thesis (Ph.D.) - Federal University of Rio Grande do Norte.

Center of Exact and Earth Sciences. Dept. of Informatics and

Applied Mathematics. Program of Post Graduation in Systems and

Computing.

1. Ideal fuzzy. 2. Prime. 3. Strongly. I. Santiago, Regivan

Hugo Nunes. II. Título.

RN/UF/CCET CDU 004:510.6

Abstract

The main aim of this investigation is to propose the notion of uniform and strong prime-

ness in fuzzy environment. First, it is proposed and investigated the concept of fuzzy

strongly prime and fuzzy uniformly strongly prime ideal. As an additional tool, the

concept of t/m systems for fuzzy environment gives an alternative way to deal with

primeness in fuzzy. Second, a fuzzy version of correspondence theorem and the radical

of a fuzzy ideal are proposed. Finally, it is proposed a new concept of prime ideal for

Quantales which enable us to deal with primeness in a noncommutative setting.

iii

Dedication

This investigation is dedicated to you, the reader, in hopes that you will ﬁnd what you

are looking for.

Acknowledgments

This work would not have been possible without the support of some people. Many

thanks to my supervisor, Regivan H. N. Santiago, who read my numerous revisions and

helped me to make some sense of the confusion. Also, thanks to my committee members,

Benjamín R. C. Bedregal, Elaine Gouvêa Pimentel, Laércio Carvalho de Barros, Edward

Hermann Haeusler, and João Marcos de Almeida. Thanks to the University of Rio

Grande do Norte for providing me with a way to learn and walk in the research path.

Thanks to Southwest Bahia State University for providing me with the ﬁnancial means

to complete this project. Thanks to my friends Adriano Dodó and Ronildo P. A. Moura

for discussing some clever ideas. Special thanks to Eliane S. A. Bergamaschi always

oﬀering support and love.

Contents

1 Introduction 1

1.1 Historical Facts ................................. 1

1.2 The Main Problem ............................... 2

2 Rings and Ideals 4

2.1 Prime, Strongly Prime and Uniformly Strongly Prime Rings ........ 6

2.2 Strongly Prime and Uniformly Strongly Prime Ideals ............ 9

3 Strongly Prime Fuzzy Ideals 12

3.1 Theory of fuzzy ideals ............................. 12

3.1.1 Fuzzy Subrings and Fuzzy Ideals ................... 12

3.1.2 Fuzzy Prime Ideals ........................... 14

3.1.3 Fuzzy Maximal Ideals ......................... 14

3.2 Strongly Prime Fuzzy Ideals .......................... 14

3.3 Strongly Prime Radical of a Fuzzy Ideal ................... 16

3.4 Semi-Strongly Prime and Strongly Primary Ideals .............. 19

3.5 Special Strongly Prime Fuzzy Ideals ..................... 21

4 Uniformly Strongly Prime Fuzzy Ideals 23

4.1 Introduction ................................... 23

4.2 Extra Results on Uspf Ideals ......................... 27

4.3 US Fuzzy Radical ................................ 29

4.3.1 Extra Results on US Radical ..................... 31

4.4 The Fuzzy m- and t-systems .......................... 31

4.4.1 The Fuzzy m- and t-Systems ..................... 32

5 Some Properties of Fuzzy Ideals 35

5.1 Preliminaries .................................. 35

5.2 The Isomorphism ................................ 37

5.3 An Equivalence of Fuzzy Ideals ........................ 39

5.4 Final Remarks ................................. 43

6 Prime Ideals and Fuzzy Prime Ideals Over Noncommutative Quantales 44

6.1 Introduction ................................... 44

6.2 Primeness in Quantales ............................ 45

6.3 Strong Primeness in Quantales ........................ 50

6.4 Fuzzy Prime and Fuzzy usp Ideals in Quantales ............... 52

6.5 Final Remarks ................................. 55

7 Next Steps 56

A Publications 58

A.1 Published Studies ................................ 58

A.2 Unpublished Studies .............................. 60

Bibliography 62

vii

List of Symbols

Anrright annihilator

Anlleft annihilator

Fxinsulator of x

[a]the set {x∈R:x≡a}

R\Ithe complement of the I in R

R/I the quotient ring by I

xF the set xF ={xf :f∈F}

F x the set F x ={fx :f∈F}

xF y the set xF y ={xfy :f∈F}

sp(S)the set of all strongly prime ideals of S

spf(S)the set of all strongly prime ideals of S that contains the Ker(f)

SP (S)the set of all strongly prime fuzzy ideals of S

SPf(S)the set of all strongly prime fuzzy ideals where Iα⊂spf(S)

Ker(f)the kernel of f

µIthe membership function of the I

I(x)equivalent to µI(x)

f(I)αthe α-cut of the fuzzy set f(I)

VFminimum of the set F

WFmaximum of the set F

x∧yminimum of the set {x, y}

x∨ymaximum of the set {x, y}

r+Ileft coset

√Istrongly radical or Levitzki radical

√Iuniformly strongly radical of a fuzzy ideal

US(I)uniformly strongly radical

I∗the set {x∈R:I(x) = I(0)}

Iαthe α-cut or α-level, {x∈R:I(x)≥α}

ASSP almost special strongly prime ideal

SSP special strongly prime ideal

usp uniformly strongly prime ideal

uspf uniformly strongly prime fuzzy ideal

viii

List of Symbols

2Zthe set {2q:q∈Z}

F IF V (R)the set of all ﬁnite-valued fuzzy ideals of R

LF I (R)the set of all fuzzy ideals of R

Im(µ)image of the membership µ

⊥least element

>greatest

hAiideal generated by Ain a quantale

Chapter 1

Introduction

1.1 Historical Facts

In 1871, Dedekind generalized the concept of prime numbers to prime ideals, which were

deﬁned in a similar way, namely, as a proper subset of integers that contains a product

of two elements if and only if it contains one of them. For example, the set of integers

divisible by a ﬁxed prime pform a prime ideal in the ring of integers. Also, the integer

decomposition into the product of powers of primes has an analogue in rings. We can

replace prime numbers with prime ideals, as long as powers of prime integers are not

replaced by powers of prime ideals but by primary ideals. The uniqueness of the latter

decomposition was proved in 1915 by Macaulay. Thus, we can think about of prime

ideals as atoms, like prime numbers are atoms in the ring of integers.

From the properties of the integers we can develop a general structure called ring. Then,

the concept of primeness may be extended to a commutative ring in a certain way, for

example: a prime ring Ris a ring where the (0) zero ideal is prime that is, given aand b

nonzero elements in R, there exists r∈R, such that arb is nonzero in R. In commutative

ring theory prime rings are integral domains, i.e. rings where ab = 0 implies a= 0 or

b= 0. Suppose that, given a6= 0 in Rthere exists a ﬁnite nonempty subset Fa⊆R,

such that afr = 0 implies r= 0, for all f∈Fa. In that case we have a strong primeness

condition on a ring. If the ring satisﬁes this last condition it is called strongly prime ring

(shortened sp ring). If the same Fcan be chosen for any nonzero element in R, then the

ring Ris called uniformly strongly prime (usp) ring.

Strongly prime rings were introduced in 1974, as a prime ring with ﬁnite condition in

the generalization of results on group rings proved by Lawrence in his PhD.’s thesis [1].

In 1975, Lawrence and Handelman [2] came up with properties of these rings and proved

Chapter 1. Introduction 2

important results, showing that all prime rings can be embedded in an sp ring; and that

all sp rings are nonsingular.

After that, in 1987, Olson [3] published a relevant paper about usp rings and usp radical.

He proved that the usp rings generate a radical class which properly contains both the

right and left sp radicals and which is not contained in Jacobson and Brown-McCoy

radicals.

In 1965, Zadeh [4] introduced fuzzy sets and in 1971, Rosenfeld [5] introduced fuzzy sets

in the realm of group theory and formulated the concept of fuzzy subgroups of a group.

Since then, many researchers have been engaged in extending the concepts/results of

abstract algebra to the broader framework of the fuzzy setting. Thus, in 1982, Liu [6]

deﬁned and studied fuzzy subrings as well as fuzzy ideals. Subsequently, Liu himself

(see [7]), Mukherjee and Sen [8], Swamy and Swamy [9], and Zhang Yue [10], among

others, fuzziﬁed certain standard concepts/results on rings and ideals. For example:

Mukherjee was the ﬁrst to study the notion of prime ideal in a fuzzy setting. Those

studies were further carried out by Kumar in [11] and [12], where the notion of nil

radical and semiprimeness were introduced.

After Mukherjee’s deﬁnition of prime ideals in the fuzzy setting, many investigations

extended crisp (classic) results to fuzzy setting. But Mukherjee’s deﬁnition was not

appropriate to deal with noncommutative rings. In 2012, Navarro, Cortadellas and

Lobillo [13] drew attention to this speciﬁc problem. They proposed a new deﬁnition of

primeness for fuzzy ideals for noncommutative rings holding the idea of “fuzziﬁcation” of

primeness introduced by Kumbhojkar and Bapat [14,15] to commutative rings, which is

coherence with α- cuts. Thus, Navarro et. al. [13] reopened the possibility of developing

fuzzy results for general rings of prime ideals.

1.2 The Main Problem

As it is known, after the Lawrence and Handelman’s paper many researchers developed

results about strong/uniform primeness (see [16–19]), but nothing was made in fuzzy and

quantale setting. Hence, this is the question: Could we have the strong/uniform prime-

ness in fuzzy and quantale setting? This thesis attempts to ﬁll this gap by proposing/in-

vestigating this concept in both environments. Therefore, motivated by translating the

concept of strong primeness for fuzzy setting, I decided to build a deﬁnition of strongly

prime fuzzy ideal in which the ﬁrst attempt was based on α-cuts (chapter 2and pub-

lished in [20]). In this approach every result for fuzzy environment has its counterpart

in a classical crisp setting. Afterwards, I realized that all proofs were based on α-cuts

Chapter 1. Introduction 3

and the results were only translated to fuzzy setting. Then, a new deﬁnition of strongly

prime fuzzy ideal was required, and should not be based on α-cuts, yet compatible in a

certain way. Thus, the second attempt was to propose such deﬁnition introduced in [21]

which can be found in the section 3.5. In this approach, we have the coherency with

α-cuts in “only one side”; namely: if a fuzzy ideal is strongly prime, then all α-cuts are

crisp strongly prime ideals. But the converse of this statement is still open up to now.

Instead of the concept of strongly prime ideal, the concept of uniformly strongly prime

ideal is more suitable to translate to fuzzy setting. Thus, in [22] (chapter 3) I introduce

the uniform fuzzy concept, compatible with α-cuts. In this approach I rediscovered

some crisp results on uniformly strongly prime (usp) ideals for fuzzy setting and the

compatibility with Navarro’s deﬁnition of fuzzy prime ideals. For example it is shown

that a fuzzy usp ideal is a fuzzy prime ideal without using α-cuts to prove this statement.

Also, some crisp results are no longer valid in fuzzy setting. For example, in crisp setting,

an ideal Iis usp iﬀ the quotient ring R/I is a usp ring, but as you shall soon see (example

7) this is not true in fuzzy setting.

When I began to study primeness in quantales setting I realized that some authors (see

Chapter 7) were working on noncommutative quantale with an elementwise deﬁnition

for prime ideals. As it is known, in noncommutative ring theory, prime ideals are deﬁned

based on ideals instead of on elements. Thus, I ﬁrstly decided to provide a concept

of prime ideal for a general (commutative and noncommutative) quantale in which the

elementwise prime ideal deﬁnition was replaced by another based on ideals over the

quantale. Hence, it was required to develop a crisp study for prime ideals before starting

the investigation of sp/usp ideals for quantales.

This thesis is organized as follows: Chapter 2provides an overview about the ring and

fuzzy ring theory. It also contains the deﬁnition and results of sp/usp rings and ideals

in a crisp setting; Chapter 3contains the results discovered by the authors in [20] about

fuzzy sp ideals; Chapter 4introduces the usp fuzzy ideal and its radical, all results in this

chapter are based on [22,23], except the unpublished section 4.4 introduces a new tool

for dealing with prime fuzzy ideals and usp fuzzy ideals called systems, where may extend

the Navarro’s paper, since the complement of fuzzy prime ideal is a fuzzy system (see

corollary 23); Chapter 5shows some extra results on fuzzy ideals; Chapter 6introduces

a new concept of prime ideal in quantales; Chapter 7contains some thoughts about the

next studies; Finally, Appendix Acontains published and unpublished studies.

Chapter 2

Rings and Ideals

This chapter introduces some deﬁnitions and results that will be required in this inves-

tigation. Here, we start by deﬁning prime rings/ideals and uniformly strongly prime

rings/ideals.

Deﬁnition 1.Aring is a nonempty set Rof elements closed under two binary operations

+ and ·with the following properties:

(i) (R,+) (that is, the set Rconsidered with the single operation of addition) is an

abelian group (whose identity element is denoted 0R, or just 0);

(ii) The operation ·is associative: (a·b)·c=a·(b·c)for every a, b, c ∈R. Thus, (R, ·)

is a semigroup;

(iii). The operations +and ·satisfy the two distributive laws: (a+b)·c=a·c+b·c

and a·(b+c) = a·b+a·c, for every a, b, c ∈R.

If Ris a ring and there exists an element 1such that a·1 = afor every a∈Rwe say

that the ring has multiplicative identity. Also, if a·b=b·afor a, b ∈Rwe call Ra

commutative ring.

Very often we omit writing the ·for multiplication, that is, we write ab to mean a·b.

Note that there can only be one additive identity in R(because (R,+) is a group, and

a group can only have one additive identity). Also, there can be only one multiplicative

identity in R. If Ris commutative and for any a, b ∈R,ab = 0 implies a= 0 or b= 0

we call Ran integral domain. Note that the ring of n×nmatrices with integers entries

is a noncommutative ring and nor an integral domain.

Deﬁnition 2.Let Rbe a ring. A nonempty subset Iof Ris called a right ideal of Rif:

Chapter 1. Introduction 5

(a) a, b ∈Iimplies a+b∈I;

(b) given r∈R, a ∈I, then ar ∈I(that is, a right ideal absorbs right multiplication by

the elements of the ring).

Similarly we can deﬁne left ideal replacing (b) by: (b’) given r∈R, a ∈I, then ra ∈I.

If Iis both right and left ideal of R, we call Iatwo-sided ideal or simply an ideal.

For the next deﬁnition consider the following notation I·Jand xRy in which:

I·J=IJ ={i1j1+· ·· +injn:ik∈I and jk∈J, k = 1, . . . , n;where n ∈Z+}

xRy ={xry :r∈R}.

Deﬁnition 3.Aprime ideal in an arbitrary ring Ris any proper (P⊆Rand P6=R)

ideal Psuch that, whenever I, J are ideals of Rwith I J ⊆P, either I⊆Por J⊆P.

Theorem 1.[[24], Proposition 10.2] An ideal Pof a ring Ris prime iﬀ for x, y ∈R,

xRy ⊆Pimplies x∈Por y∈P.

Deﬁnition 4.An ideal Pof a ring Ris called completely prime if given aand btwo

elements of Rsuch that their product ab ∈P, then a∈Por b∈P.

Given a ring Rand a∈R, the set (a) = RaR ={x1ay1+···+xnayn:n∈N, xi, yi∈R}

is an ideal and is called the ideal generated by a.

For arbitrary rings, completely prime implies prime, but the converse is not true as we

can see in the following example:

Example 1.[13] Let (0) as an ideal generated by 0, and let Rbe the ring of 2×2ma-

trices over the real numbers. Let us show that the (0) (zero ideal) is prime, but (0) is

not completely prime by using the theorem 1. Thus, suppose that X= a b

c d !and

Y= e f

g h !are two matrices such that XRY ⊆(0). Hence X T Y = 0 0

0 0 !for

any other matrix T∈R. Let T= 1 0

0 0 !. Then

X 1 0

0 0 !Y= a b

c d ! 1 0

0 0 ! e f

g h != ae af

ce cf != 0 ⇔a=c=

0or e=f= 0,

Chapter 1. Introduction 6

X 0 1

0 0 !Y= a b

c d ! 0 1

0 0 ! e f

g h != ag ah

cg ch != 0 ⇔a=c=

0or g=h= 0,

X 0 0

1 0 !Y= a b

c d ! 0 0

1 0 ! e f

g h != be bf

de df != 0 ⇔b=d=

0or e=f= 0,

X 0 0

0 1 !Y= a b

c d ! 0 0

0 1 ! e f

g h != bg bh

dg dh != 0 ⇔b=d=

0or g=h= 0,

Hence, it should be the case that X= 0 0

0 0 !or Y= 0 0

0 0 !. Therefore X∈(0)

or Y∈(0) and then (0) is prime. Nevertheless, (0) is not completely prime, since

0 1

0 0 ! 0 1

0 0 != 0 0

0 0 !although 0 1

0 0 !/∈(0).

2.1 Prime, Strongly Prime and Uniformly Strongly Prime

Rings

A ring is called a simple ring if is a nonzero ring that has no two-sided ideals besides the

zero ideal and itself. In 1973, Formanek [25] proved that if Dis a integral domain and

Gcan be factored into a free product of a groups, then the group ring DG is a simple

ring. In the same year, Lawrence in his Master’s thesis showed that a generalization of

Formanek’s result was possible, in which the integral domain is replaced by a prime ring

with a ﬁniteness condition called strong primeness. Although the condition of strong

primeness was already used in a speciﬁc problem for group rings, the theory of strong

primeness became more itself interesting. As a consequence, in 1975, Lawrence and

Handelman [2] began to study the strongly prime rings for which some results were

discovered, for example that every prime ring may be embedded in a strongly prime ring

and that the Artinian strongly prime rings have a minimal right ideal.

Deﬁnition 5.A ring Ris prime if for any two elements aand bof R,arb = 0 for all rin

Rimplies that either a= 0 or b= 0.

Chapter 1. Introduction 7

We can think of prime rings as a simultaneous generalization of both integral domains

and simple rings. In the commutative case Ris prime iﬀ Ris an integral domain.

Deﬁnition 6.Let Abe a subset of a ring R. The right annihilator of Ais deﬁned as

Anr(A) = {x∈R:Ax = (0)}. Similarly, we can deﬁne the left annihilator Anl

Deﬁnition 7.[2] A ring Ris called right strongly prime if for each nonzero x∈Rthere

exists a ﬁnite nonempty subset Fxof Rsuch that the Anr(xFx) = (0).

When Ris right strongly prime we can prove that Fxis unique and called a right insulator

for x.

Parmenter, Stewart and Wiegandt [26] have shown that the deﬁnition of right strongly

prime is equivalent to:

Proposition 1.A ring Ris right strongly prime if each nonzero ideal Iof Rcontains a

ﬁnite subset Fwhich has right annihilator zero.

It is clear that every right strongly prime ring is a prime ring. It is also possible to deﬁne

left strongly prime in a manner analogous to that for right strong primeness. Handelman

and Lawrence showed that these two concepts are distinct, by building a ring that is right

strongly prime but not left strongly prime (see [2], Example 1).

Example 2.If Iis an ideal in a simple ring R, then I= (0) or I=R. Thus, if I6= (0),

then I=R. Let F={1}then Anr(F) = {0}. Hence, according to deﬁnition 1Ris a

right strongly prime ring.

Example 3.Adivision ring is a nonzero ring such that multiplicative identity in which

every nonzero element ahas a multiplicative inverse, i.e., an element xwith ax =xa = 1.

It is easy to see that a division ring is a simple ring. Therefore, it is a right strongly

prime ring.

Aﬁeld is a commutative division ring with multiplicative identity. Therefore strongly

prime ring.

Example 4.Consider Znthe commutative ring of integers mod n, for n > 1. If a∈Z,

the class of ais [a] = {x∈Z: (x mod n) = a}. Note that if nis not a prime number,

then there exists p, q ∈Zsuch that n=pq, where 0<p<nand 0< q < n. Hence,

[pq]=0in Zn, but [p]6= 0 and [q]6= 0. We conclude that Znis not a integral domain

and as a consequence Znis not a prime ring. Thus, Znis not right strongly prime ring.

On the other hand, if nis prime, Znis a ﬁeld, hence right strongly prime ring.

Deﬁnition 8.A ring is a bounded right strongly prime ring of bound n, if each nonzero

element has an insulator containing no more than nelements and at least one element

has no insulator with fewer than nelements.

Chapter 1. Introduction 8

Deﬁnition 9.A ring is called uniformly right strongly prime if the same right insulator

may be chosen for each nonzero element.

Since an insulator must be ﬁnite, it is clear that every uniformly strongly prime ring is a

bounded right strongly prime ring of bound n. Again, analogous deﬁnitions of bounded

left strongly prime and uniformly left strongly prime can be formulated. As in the case

with the notation of strong primeness it is possible to ﬁnd rings which are bounded left

strongly prime but not bouded right strongly prime, and vice-versa (see [2], Example 1).

However, Olson [3] showed that the concept of uniformly strongly prime ring is two-sided

due to the following result:

Lemma 2.[3] A ring Ris right/left uniformly strongly prime iﬀ there exists a ﬁnite

subset F⊆Rsuch that for any two nonzero elements xand yof R, there exists f∈F

such that xfy 6= 0.

Corollary 3.[3]Ris uniformly right strongly prime ring if and only if Ris uniformly left

strongly prime ring.

Lemma 4.[3] The following are equivalent:

i) Ris a uniformly strongly prime ring;

ii) There exists a ﬁnite subset F⊆Rsuch that xF y = 0 implies x= 0 or y= 0, where

x, y ∈R;

iii) For every a6= 0, a ∈R, there exists a ﬁnite set F⊂(a)such that xF y = 0 implies

x= 0 or y= 0, where x, y ∈R;

iv) For every a6= 0, a ∈R, there exists a ﬁnite set F⊂(a)such that xF x = 0 implies

x= 0, where x∈R;

v) For every ideal I6= 0, there exists a ﬁnite set F⊂Isuch that xF y = 0 implies x= 0

or y= 0, where x, y ∈R;

vi) For every ideal I6= 0, there exists a ﬁnite set F⊂Isuch that xF x = 0 implies x= 0,

where x∈R;

vii) For every a6= 0, a ∈R, there exists a ﬁnite set F⊂Rsuch that xF aF x = 0 implies

x= 0, where x∈R;

viii) For every a6= 0, a ∈R, there exists a ﬁnite set F⊂Rsuch that xF aF y = 0 implies

x= 0 or y= 0, where x, y ∈R.

Chapter 1. Introduction 9

2.2 Strongly Prime and Uniformly Strongly Prime Ideals

Let Ibe a two-sided ideal in R. We may deﬁne an equivalence relation ∼Ion Ras

follows: a∼biﬀ b−a∈I. In case a∼b, we say that aand bare congruent modulo I.

The equivalence class of the element ain Ris given by:

[a] = a+I={a+r:r∈I}.

The set of all such equivalence classes is denoted by R/I and it is a ring called the

quotient ring, where the operations are:

[a]+[b]=(a+I)+(b+I) = (a+b) + I= [a+b];

[a]·[b] = (a+I)(b+I) = (ab) + I= [ab].

The zero-element of R/I is [0] = 0 + I=I.

From this point forward, strongly prime means right strongly prime.

Deﬁnition 10.[2] An ideal Iin a ring Ris strongly prime if R/I is a strongly prime ring.

Proposition 2 ([27], Proposition 4.3, Chapter IX).An ideal Iof a ring is prime iﬀ R/I

is a prime ring.

We reproved the following result.

Proposition 3.The ideal Iis a strongly prime ideal in Riﬀ for every x∈R\Ithere

exists a ﬁnite subset Fxof Rsuch that if xFxr⊆Iimplies r∈I.

Proof. Let Istrongly prime ideal and R/I strongly prime ring. If x∈R\I, then

(x+I)6=Iin R/I. The insulator of x+Iin R/I is a ﬁnite set F∗

x={f1+I,...,fk+I}

for some particular choice of the fi. Let Fx={f1, . . . , fn} ⊆ Rbe such that xFxr⊆I.

Hence, (xF ∗

xr+I) = Iin R/I. Thus, (r+I) = Iand r∈I. Conversely, given (x+I)6=I

in R/I there exists a ﬁnite set Fx⊂Rsuch that (x+I)(Fx+I)(y+I)=(xFxy+I) = I.

This implies xFxy∈Iand y∈I. Thus, (y+I) = Iand R/I is a strongly prime ring.

Corollary 5.Pis strongly prime ideal in Riﬀ for every x, y ∈R, if xP y ⊆Pand xy ∈P,

then either x∈Por y∈P.

Proposition 4.[2] Let Pbe a proper ideal of a ring R. The following conditions are

equivalent:

(i) Pis strongly prime.

(ii) For every ideal I⊃Pthere exists a ﬁnite set F⊆Isuch that if F a ⊆P, then a∈P.

Chapter 1. Introduction 10

Proposition 5.[2] If Ris a ﬁnite ring, then every prime ideal is a strongly prime ideal.

A ring homomorphism is a function between two rings which respects the structure. More

explicitly, if Rand Sare rings, then a ring homomorphism is a function f:R−→ S

such that: f(a+b) = f(a) + f(b),f(ab) = f(a)f(b)for all aand bin Rand f(1R)=1S.

Let f:R−→ Sbe a homomorphism of rings. Let sp(S)be the set of all strongly prime

ideals of Sand spf(R) = {I∈sp(R) : I⊇Ker(f)}where Ker(f) = {r∈R:f(r) =

0}. According to the isomorphism theorem for rings, if fis a epimorphism, there is a

bijection between spf(R)and sp(S). In the next chapter we will show its counterpart in

a fuzzy setting (see proposition 12).

Theorem 6.[28] Let f:R−→ Sbe an epimorphism of rings. Then

(i) f(I)∈sp(S)for any I∈spf(R);

(ii) f−1(I)∈spf(R)for any I∈sp(S);

(iii) Deﬁne the mapping Ψ : spf(R)−→ sp(S),Ψ(I) = f(I). Then Ψis a bijection.

Proposition 6.Let f:R−→ Sbe an isomorphism of rings.

a) P⊆Ris a prime ideal iﬀ f(P)is a prime ideal of S.

b) P⊆Ris a strongly prime ideal iﬀ f(P)is a strongly prime ideal of S.

Proposition 7 ([29]).Let Rbe a ring and A, B be ideals of Rwith A⊆B: If Bis strongly

prime, then there exists a minimal element in S={P⊆R:P is strongly prime ideal and A ⊆

P⊆B}.

Deﬁnition 11.A proper ideal Iof a ring Ris a uniformly strongly prime ideal if R/I is

a uniformly strongly prime ring.

We reproved the following two results, because they provide another characterization of

uniformly strongly prime ideals.

Proposition 8.An ideal Iof a ring Ris uniformly strongly prime iﬀ there exists a ﬁnite

set F⊆Rsuch that xF y ⊆Iimplies x∈Ior y∈I, where x, y ∈R.

Proof. Let Ibe a uniformly strongly prime ideal of the ring R. Then R/I is uniformly

strongly prime ring. Let F∗={f1+I,...,fk+I}be a insulator for R/I for some

particular choice of the fiand F={f1, . . . , fk}. Choose x, y ∈Rsuch that xF y ⊆I.

Hence (x+I)F∗(y+I) = I. By hypothesis (x+I) = Ior (y+I) = I. Thus, x∈Ior y∈I.

Chapter 1. Introduction 11

Conversely, let (x+I),(y+I)∈R/I. Suppose (x+I)(F+I)(y+I) = xF y +I=I.

Hence, xF y ⊆I, by hypothesis x∈Ior y∈I. Thus, (x+I) = Ior (y+I) = I.

Therefore, R/I is a uniformly strongly prime ring.

Proposition 9.An ideal Iof a ring Ris uniformly strongly prime iﬀ there exists a ﬁnite

set F⊆Rsuch that for any two nonzero elements xand yof R\I(the complement of

Iin R), there exists f∈Fsuch that xfy /∈I.

Proof. Let Ibe a uniformly strongly prime ideal. Then, R/I is uniformly strongly

prime. Let {f1+I,...,fk+I}be a insulator for R/I for some particular choice of

the fi. Choose x, y ∈R\I. Then (x+I)and (y+I)are nonzero elements in R/I

and according to lemma 2there exists fi+I∈F∗for some i= 1, . . . , k such that

(x+I)(fi+I)(y+I) = xfiy+I6=I. Then xfiy∈R\I. Conversely, if (x+I)and

(y+I)are nonzero elements of R/I then xand yare in R\I. By hypothesis there exists

f∈Fsuch that xfy ∈R\I. That is (x+I)(f+I)(y+I) = xf y +I6=I. According to

lemma 2R/I is a uniformly strongly prime ring and {f+I:f∈F}is a insulator for

R/I.

Chapter 3

Strongly Prime Fuzzy Ideals

In this chapter the concept of strongly prime fuzzy ideal for rings is deﬁned. Also,

it is shown that the Zadeh’s extension of homomorphism somewhat preserves strong

primeness and that every strongly prime fuzzy ideal is a prime fuzzy ideal as well as

every fuzzy maximal is a strongly prime fuzzy ideal. The concept of strongly prime

radical of a fuzzy ideal and its properties are investigated. It is proved that Zadeh’s

extension preserves strongly prime radicals. A version of theorem of correspondence for

strongly prime fuzzy ideals is also showed. Besides, we propose new algebraic fuzzy

structures, namely: strongly primary,strong radical,Special Strongly Prime (SSP) and

Almost Special Strongly Prime (ASSP). At the end of this chapter it is shown the relation

between strong primary and strong radicals as well as the connection between the classes

SP, SSP and ASSP. All results in this chapter can be found in [20,21,30].

For fuzzy ideals and Prime fuzzy ideals we recommend ﬁrst of all [13] and then [6–10].

3.1 Theory of fuzzy ideals

3.1.1 Fuzzy Subrings and Fuzzy Ideals

By a fuzzy set we mean the classical concept deﬁned in [4], that is, a fuzzy set over a

base set Xis a set map µ:X−→ [0,1]. The intersection and union of fuzzy sets is

given by the point-by-point inﬁmum and supremum. We shall use the symbols ∧and ∨

for denoting the inﬁmum and supremum of a collection of real numbers.

Deﬁnition 12.A fuzzy subset I:R−→ [0,1] of a ring Ris called a fuzzy subring of R

if, for all x, y ∈R: the following requirements are met:

1) I(x−y)≥I(x)∧I(y);

Chapter 3. Strongly Fuzzy Primeness 13

2) I(xy)≥I(x)∧I(y);

If condition 2) is replaced by I(xy)≥I(x)∨I(y), then Iis called a fuzzy ideal of R.

Note that if Iis a fuzzy ideal of a ring R, then I(1) ≤I(x)≤I(0) for all x∈R.

Deﬁnition 13.Let µbe any fuzzy subset of a set Sand let α∈[0,1]. The set {x∈

S:µ(x)≥α}is called a α-cut of µwhich is symbolized by µα.

Clearly, if t>s, then µt⊆µs.

Proposition 10.[31] A fuzzy subset Iof a ring Ris a fuzzy subring/fuzzy ideal of Riﬀ

all α-cuts Iαare subrings/ideals of R.

Here is an example of a fuzzy subring of a ring Rwhich is not a fuzzy ideal of R.

Example 5.Let Rdenote the ring of real numbers under the usual operations of addition

and multiplication. Deﬁne a fuzzy subset µof Rby

µ(x) = (t, if x is rational,

t0, if x is irrational,

where t, t0∈[0,1] and t>t0. Note that µt=Qand µt0=R. Thus µtis a subring

according to the Proposition 10, but not a fuzzy ideal.

Deﬁnition 14 (Zadeh’s Extension).[4] Let fbe a function from set Xinto Y, and let µ

be a fuzzy subset of X. The Zadeh extension of fis the fuzzy subset f(µ)of Y, where

the membership function is: For all y∈Y,

f(µ)(y) = 









∨{µ(x) : x∈X, f (x) = y},if f−1(y)6=∅

0,otherwise.

If λis a fuzzy subset of Y, we deﬁne the fuzzy subset of X, denoted as f−1(λ), where

f−1(λ)(x) = (λ◦f)(x).

Proposition 11.[32] If f:R−→ Sis a ring homomorphism and I:R−→ [0,1] and

J:S−→ [0,1] are fuzzy ideals, then

i) f−1(J)(according to the last deﬁnition) is a fuzzy ideal which is constant on Ker(f)

(Kernel of f);

ii) f−1(Jα) = f−1(J)α, where α=J(0);

iii) If fis an epimorphism, then f(I)is a fuzzy ideal and f f −1(J) = Jand f(Iα) = f(I)α,

where α=I(0);

iv) If Iis constant on Ker(f), then f−1f(I) = I.

Chapter 3. Strongly Fuzzy Primeness 14

3.1.2 Fuzzy Prime Ideals

Deﬁnition 15.[13] Let Rbe a ring with unity. A non-constant fuzzy ideal P:R−→ [0,1]

is said to be prime or fuzzy prime ideal if for any x, y ∈R,^P(xRy) = P(x)∨P(y).

Proposition 12.[13] Let Rbe an arbitrary ring with unity and P:R−→ [0,1] be a

non-constant fuzzy ideal of R. The following conditions are equivalent:

(i) Pis prime;

(ii) Pαis prime for all P(1) < α ≤P(0);

(iii) R/Pαis a prime ring for all P(1) < α ≤P(0);

(iv) For any fuzzy ideal J, if J(xry)≤P(xry)for all r∈R, then J(x)≤P(x)or

J(y)≤P(y).

Note that if Pis a fuzzy ideal, then P(xry)≥P(x)∨P(r)∨P(y)≥P(x)∨P(y)for all

r∈R. Thus, ^P(xRy)≥P(x)∨P(y).

Deﬁnition 16.Let Ibe a fuzzy ideal of a ring R. For all r∈Rdeﬁne fuzzy left coset

r+I, where (r+I)(x) = I(x−r).

The deﬁnition above allow us to built the quotient ring R/I in the same way as we did

in crisp setting.

3.1.3 Fuzzy Maximal Ideals

Deﬁnition 17.[33] Let Mbe a fuzzy ideal of a ring R. Then Mis called fuzzy maximal

of Rif the following conditions are met:

(i) Mis non-constant;

(ii) for any fuzzy ideal νof R, if M⊆νthen either M∗=ν∗or ν=µR, where

M∗={x∈R:M(x) = M(0)},ν∗={x∈R:ν(x) = ν(0)}and µR(x) = 1 if x∈R

and µR(x)=0otherwise.

Proposition 13.[33] Let Mbe a fuzzy maximal ideal of a ring R. Then M(0) = 1.

Proposition 14.[33] Let Mbe a fuzzy maximal ideal of a ring R. Then |Im(M)|= 2.

3.2 Strongly Prime Fuzzy Ideals

In this section, the notion of strongly prime fuzzy ideal is introduced and the well-known

crisp results in the fuzzy setting are proved.

Chapter 3. Strongly Fuzzy Primeness 15

Deﬁnition 18.(Strongly prime fuzzy ideal) Let Rbe an arbitrary ring with unity. A

non-constant fuzzy ideal P:R−→ [0,1] is said to be strongly prime if Pαis strongly

prime for any P(1) < α ≤P(0).

Theorem 7.Every strongly prime fuzzy is prime fuzzy.

Proof. Let Pbe strongly prime fuzzy, then Pαis strongly prime for all P(1) < α ≤P(0).

Hence Pαis prime. Based on Proposition 12 Pis prime fuzzy.

Theorem 8.Let Rbe a ﬁnite ring with unity. Pis a strongly prime fuzzy iﬀ Pis prime

fuzzy.

Proof. Immediately from Proposition 5, deﬁnition 18 and Proposition 12.

The next two results show that Zadeh’s extension preserves prime fuzzy and strongly

prime fuzzy when fis an isomorphism.

Proposition 15.Let f:R−→ Sbe an isomorphism of rings. If Pis a prime fuzzy ideal

of R, then f(P)is a prime fuzzy ideal of S.

Proof. Since fis bijective, given y∈S, there is a unique x∈Rsuch that f(x) = y.

Hence, f(P)(y) = P(x)and f(x) = yfor all y∈S. Then:

f(P)α={y∈S:f(P)(y)≥α}

={f(x)∈S:P(x)≥α}

=f(Pα).

As Pis prime fuzzy, by Proposition 12,Pαis prime P(1) < α ≤P(0) and by Proposition

6f(Pα)is prime and then f(P)αis prime for all P(1) < α ≤P(0). By Proposition 12

once more f(P)is prime fuzzy.

Theorem 9.Let f:R−→ Sbe an isomorphism of rings. If Pis a strongly prime fuzzy

ideal of R, then f(P)is a strongly prime fuzzy ideal of S.

Proof. Similar to demonstration of Proposition 15

Proposition 16.Any strongly prime fuzzy ideal contains a minimal strongly prime fuzzy

ideal.

Proof. Let Pbe a strongly prime fuzzy ideal over a ring R. Then, P∗is strongly prime

and by Proposition 7it has a minimal strongly prime M⊆P∗. Deﬁne

ν(x) = (P(0) if x ∈M

P(1) otherwise.

Chapter 3. Strongly Fuzzy Primeness 16

As P(0) 6=P(1),ναis strongly prime for all α∈[0,1]. Thus, νis equivalent to the

characteristic map of Mand ν⊆P.

Proposition 17.Any strongly prime fuzzy ideal contains properly another strongly prime

fuzzy ideal.

Proof. Let Pbe a strongly prime fuzzy. Consider the fuzzy set ν=1

2·P⊂Pdeﬁned

by ν(x) = 1

2P(x). Both fuzzy sets share the same level subsets. So νis a strongly prime

fuzzy ideal.

Theorem 10.Let Rbe a ring with unity. Any maximal fuzzy ideal is a fuzzy strongly

prime ideal.

Proof. Let Mbe a maximal fuzzy ideal. By Proposition 13 and 14 Im(M) = {M(1),1},

M(0) = 1 and M∗is a crisp maximal. Let M(1) < α ≤M(0) then α=M(0). Thus,

Mα=M∗is a crisp maximal ideal. By crisp theory, every maximal ideal is strongly

prime, and then Mαis strongly prime. Therefore, Mis strongly prime fuzzy.

The converse of theorem 10 is not true as is shown by the following example.

Example 6.Let R=Zbe the ring of integers and I(x) = (1if x = 0

0otherwise . Note

that Iα= (0) for all I(1) < α ≤I(0). Thus, Iis a strongly prime fuzzy ideal. Now let

ν(x) = (1if x ∈2Z

0otherwise .

Where 2Z={n∈Z:n= 2q, q ∈Z}.

Clearly I⊆ν, but ν6=Zand ν∗= 2Z6= (0) = I∗. Therefore, Iis not maximal.

3.3 Strongly Prime Radical of a Fuzzy Ideal

The right strongly prime radical of a ring Ris deﬁned to be the intersection of all

right strongly prime ideals of Rand the left strongly primeness determines the left

strongly prime radical. An example given by Parmenter, Passman and Stewart [34]

showed that these two radicals are distinct. In this section we deﬁne the concept of right

strongly radical (shortly strongly radical) of a fuzzy ideal. Also, it is shown a version

of Correspondence Theorem and a right strongly prime radical (shortly sp radical) of a

fuzzy ideal is deﬁned and investigated. Throughout this section, unless stated otherwise,

Rhas identity.

Chapter 3. Strongly Fuzzy Primeness 17

Proposition 18.Let f:R−→ Sbe a epimorphism of rings such that f−1(Y)is a ﬁnite

set for all Y⊆S. If Iis a fuzzy set of Rand Ja fuzzy set of S, then f(Iα) = f(I)αand

f−1(Jα) = f−1(J)α.

Proof. Consider f(Iα) = {y∈S:y=f(x), x ∈Iα}and f(I)α={y∈S:f(I)(y)≥

α}. Let y∈f(Iα),y=f(x0)where I(x0)≥α. Thus, f(I)(y) = sup{I(x) : f(x) =

y} ≥ I(x0)≥αand then y∈f(I)α. On the other side, let y∈f(I)α, i.e. f(I)(y) =

sup{I(x) : f(x) = y} ≥ α. As fis surjective, there exists x0∈R, where α≤I(x0)≤

sup{I(x) : f(x) = y}=f(I)(y). Thus, x0∈Iαand then f(x0) = y∈f(Iα).

To prove f−1(Jα) = f−1(J)α, let x∈f−1(Jα), then f(x)∈Jα. Thus, f−1(J)(x) =

J(f(x)) ≥αand, therefore, x∈f−1(J)α. Now let x∈f−1(J)αthen J(f(x)) =

f−1(J)(x)≥αand therefore f(x)∈Jα. In this case, it is not necessarily used f−1(Y)

as a ﬁnite set.

Theorem 11.Let f:R−→ Sbe a epimorphism of rings such that f−1(Y)is a ﬁnite set

for all Y⊆S. If Iis a sp fuzzy ideal of Rsuch that Ker(f)⊆Iαfor I(1) < α ≤I(0),

then f(I)is sp fuzzy ideal of R.

Proof. Let Ibe a sp fuzzy ideal of R, where Iα∈spf(R)for I(1) < α ≤I(0). Applying

Theorem 6, (i) f(Iα)∈sp(S). By the Proposition 18 f(I)αis sp fuzzy ideal of S. Thus,

f(I)∈sp(S).

Proposition 19.Let f:R−→ Sbe an epimorphism of rings. If Jis a sp fuzzy ideal of

S, then f−1(J)is a sp fuzzy ideal of R, where f−1(J)α⊇Ker(f)for J(1) < α ≤J(0).

Proof. It is a consequence from proposition 18 and theorem 6(ii).

For the next result, consider SPf(R) = {Iis sp fuzzy ideal of R :Iα∈spf, I (1) < α ≤

I(0)}and SP (S)is the set of all sp fuzzy ideals of S.

Theorem 12.(Correspondence Theorem) Let f:R−→ Sbe an epimorphism of rings

such that f−1(Y)is a ﬁnite set for all Y⊆S. Then, there exists a bijection between

SPf(R)and S P (S).

Proof. Deﬁne Ψ : SPf(R)−→ S P (S),Ψ(I) = f(I). Let I , M ∈SPf(R), where I6=M.

Thus, there exists x∈R, where I(x)6=M(x), if α=I(x), then Iα6=Mα. According

to proposition 18 and Theorem 6,f(I)α=f(Iα)6=f(Mα) = f(M)α. Therefore, Ψ

is injective. On the other hand, let J∈S P (S). As Jαis SP by Theorem 6, we have

Chapter 3. Strongly Fuzzy Primeness 18

f−1(Jα)∈spf(R), by Proposition 18,f−1(Jα) = f−1(J)α. Thus, f−1(J)αis SP and

f−1(J)∈SPf(R). Moreover, Ψ(f−1(J)) = f(f−1(J)) = J. Therefore, Ψis surjective.

Deﬁnition 19.Given a crisp ideal Iof a ring R, the strongly radical(or Levitzki radical)

of Iis s

√I=T{P:P⊇I, P is strongly prime}.

Deﬁnition 20.Let Ibe a fuzzy ideal of R, the strongly radical of Iis s

√I=\

P∈SI

where SIis the family of all sp fuzzy ideals Pof Rsuch that I⊆P.

Clearly s

√Iis an ideal, and if Iis a sp fuzzy ideal, then s

√I=I

Proposition 20.Let Ibe a nonconstant fuzzy ideal of ring R. Then:

i) s

√I∗⊆(s

√I)∗, where I∗={x∈R;I(x) = I(0)};

ii) s

√I(x)=1for all x∈(s

√I)∗;

iii) s

√I(0) = I(0),s

√I(1) = I(1);

iv) I∗⊆(s

√I)∗;

v) I⊆s

√I.

Proof. Straightforward.

Proposition 21.If I , J are a fuzzy ideal of a ring R, then:

(i) if I⊆J, then s

√I⊆s

√J;

(ii) s

√I=s

√I;

(iii) Iα⊆(s

√I)α;

(iv) If Iis SP fuzzy, then s

√Iα= ( s

√I)α;

(v) s

√I∩J⊆s

√I∩s

√J.

Proof. (i) s

√J=\

P∈SJ

P⊇\

P∈SI

P=s

√I. (ii) It is easy to see that s

√I⊆s

√I. On the

other side, let’s show SI⊆ Ss

√I. In fact, let P∈ SI, then P⊇Iusing (i) P=s

√P⊇s

√I.

(iii),(iv) and (v) is straightforward.

Chapter 3. Strongly Fuzzy Primeness 19

Proposition 22.Let f:R−→ Sbe a homomorphism of rings and Ia fuzzy ideal of R.

Then:

1)f(I)⊆f(s

√I)⊆s

qf(s

√I);

2) I⊆f−1(s

pf(I)).

Proof. 1) Straightforward.

2) As f(I)⊆s

pf(I), then f−1(f(I)) ⊆f−1(s

pf(I)). Thus, I⊆f−1(f(I)) ⊆f−1(s

pf(I)).

Proposition 23.Let f:R−→ Sbe a homomorphism of rings and Ia SP fuzzy ideal of

R. Then, f(s

√I)⊆s

pf(I).

Proof. As Iis SP fuzzy s

√I=I, then s

pf(I) = s

qf(s

√I). Thus, f(s

√I)⊆s

qf(s

√I) =

pf(I).

Proposition 24.Let f:R−→ Sbe an epimorphism of rings and Ia sp fuzzy ideal of R,

such that Ker(f)⊆Iαfor I(1) < α ≤I(0). Then, f(s

√I) = s

pf(I).

Proof. As Iis SP fuzzy ideal, I=s

√I,f(I) = f(s

√I). Using the theorem 11 f(I)is SP

fuzzy ideal and then f(I) = s

pf(I). Thus, f(s

√I) = s

pf(I) = s

pf(I).

Proposition 25.Let f:R−→ Sbe an epimorphism of rings and Ia fuzzy ideal of R

such that Ker(f)⊆Iαfor I(1) < α ≤I(0) and s

√Iis SP fuzzy ideal. Then, f(s

√I)is

SP fuzzy ideal of S.

Proof. Straightforward.

3.4 Semi-Strongly Prime and Strongly Primary Ideals

The aim of this section is to prove a strong prime fuzzy version (proposition 27) of the

following theorem:

“In a commutative ring, Iis a prime ideal iﬀ Iis semi-prime and primary ideal.”

Deﬁnition 21.A crisp or fuzzy ideal Iof ring Ris semi-strongly prime (or semi-sp), iﬀ

√I=I.

Chapter 3. Strongly Fuzzy Primeness 20

Proposition 26.Let f:R−→ Sbe a homomorphism of rings and Isemi-sp fuzzy ideal

of R, then f(I)is semi-sp fuzzy ideal of S.

Proof. f(I) = f(s

√I) = f

\

J∈PI

J

=\

J∈PI

f(J)⊇\

P∈Pf(I)

P=s

pf(I).

Thus, f(I) = s

pf(I).

Corollary 13.Let f:R−→ Sbe a homomorphism of rings.

1) If Iis semi-sp fuzzy ideal of R, then f(s

√I) = s

pf(I).

2) If Iis sp fuzzy ideal of R, then f(I)is sp fuzzy ideal of S.

Proof. Straightforward.

According to the classical deﬁnition, a proper ideal Iin a ring Ris said to be primary

whenever xy is an element of Iwe have x∈Ior yn∈I, for some n > 0. Moreover, the

last condition can be replaced by x∈Ior y∈√I, where √Iis the radical of Ideﬁned

by √I={r∈R:rn∈Ifor some positive integer n}.

Deﬁnition 22.(Crisp) A proper ideal Iof a ring Ris said to be strongly primary,

whenever xy ∈I, we have x∈Ior y∈s

√I.

According [35], Malik and Moderson, if a fuzzy ideal is primary, its α-cuts may not be

necessarily primary. Thus, we decided to deﬁne primary fuzzy from α-cuts as follows:

Deﬁnition 23.A non-constant fuzzy ideal Iof a ring Ris said to be primary fuzzy iﬀ its

α-cuts are primary ideals of R.

Deﬁnition 24.A non-constant fuzzy ideal Iof a ring Ris said to be strongly primary

fuzzy iﬀ its α-cuts are strongly primary ideals of R.

We observe that whenever Iis a primary ideal, then Iis strongly primary.

Proposition 27.(Crisp) Iis SP iﬀ Iis semi-strongly prime and primary ideal of R.

Proof. (⇒) Straightforward. (⇐) Suppose Iis not strongly prime. Then, there exists

an element x∈R−I, such that for every ﬁnite subset F⊆R, there exists r∈R, such

that xF r ⊆Iand r /∈I. Let F={1}, then there exists r∈R, such that xr ∈Iand

r /∈I. That contradicts the fact of I=s

√Iand strongly primary.

Chapter 3. Strongly Fuzzy Primeness 21

Corollary 14.If Iis SP fuzzy ideal, then Iis semi-strongly prime and primary fuzzy

ideal of R.

Proof. Straightforward.

Sometimes we deﬁne a fuzzy structure appealing to α-cuts. However, may all results will

be depending of them. Note that for the converse of Corollary 14 it is necessary to have

√Iα= ( s

√I)α. This fact occurs because our deﬁnition on primary was based on α-cuts.

Corollary 15.If Iis semi-strongly prime and primary fuzzy ideal of Rand s

√Iα= ( s

√I)α

for all I(1) < αI(0), then Iis SP fuzzy.

Proof. Straightforward.

Theorem 16.Iis SP fuzzy ideal iﬀ Iis semi-strongly prime and primary fuzzy ideal and

√Iα= ( s

√I)αfor all I(1) < α ≤I(0).

Proof. Straightforward.

3.5 Special Strongly Prime Fuzzy Ideals

In this section we provide two fuzzy structures which do not have correspondence in crisp

Algebra and are not buit from α-cuts.

Deﬁnition 25.(ASSP) Let Rbe an arbitrary ring with unity. A non-constant fuzzy ideal

I:R−→ [0,1] is said to be almost special strongly prime, if for every x∈Rthere exists

a subset Fxof R, such that I(r)≥VI(xFxr)for all r∈R.

Deﬁnition 26.(SSP) Let Rbe an arbitrary ring with unity. A non-constant fuzzy ideal

I:R−→ [0,1] is said to be special strongly prime, if for every x∈Rthere exists a ﬁnite

subset Fxof R, such that I(r)≥VI(xFxr)for all r∈R.

Proposition 28.If Iis SSP, then I is SP fuzzy ideal.

Proof. Let’s show that Iαis SP for I(1) < α ≤I(0). Let x∈Iα\R. As Iis SSP, there

exists a ﬁnite set Fx. Suppose xFxr⊆Iα, hence I(r)≥VI(xFxr)≥αand then r∈Iα.

Proposition 29.If Iis SSP, then Iis ASSP.

Proof. Straightforward.

Chapter 3. Strongly Fuzzy Primeness 22

Proposition 30.If Iis SP fuzzy ideal, then Iis ASSP.

Proof. Let x∈R, then x /∈Iβfor all I(x)< β ≤I(0). As Iβis strongly prime,

there exists a ﬁnite set Fβ

x⊆R, such that xF β

xr⊆Iβimplies r∈Iβfor all r∈R,

i.e. VI(xF β

xr)≥βimplies I(r)≥β. Let J={β∈[0,1] : I(x)< β ≤I(0)}and

Fx= ( [

β∈J

Fβ

x)∪ {y}, where I(y)> I(x). Consider r∈Rand t=VI(xFxr)> I (x).

Observe that x /∈Itand Itare SP, i.e. VI(xF t

xr)≥timplies I(r)≥t.

As Ft

x⊆Fx, then VI(xF t

xr)≥VI(xFxr) = tand I(r)≥t. Therefore, Iis ASSP.

Corollary 17.Let Ibe a fuzzy ideal and Im(I)is a ﬁnite set. Iis SSP iﬀ Iis SP.

Proof. Straightforward.

Question 1.In which conditions do the classes of ASSP and SSP coincide? Furthermore,

does Zadeh’s extension preserve ASSP and SSP?

Chapter 4

Uniformly Strongly Prime Fuzzy

Ideals

We proposed in section 3.2 a notion of sp ideals for the fuzzy environment. This deﬁnition

of sp fuzzy ideal, or shortened spf ideal, was based on α-cuts. In this approach we can

realize that all results for fuzzy environment have similar counterpart in classical algebra.

Although we could not ﬁnd (like Navarro in Deﬁnition 15 ) a pure fuzzy deﬁnition of

spf ideals, these ideas led them to propose the concept of uspf (uniformly strongly prime

fuzzy) ideal. Thus, as we shall see, it is possible to propose a notion of uspf ideals which is

not based on α-cuts. This approach is proposed in order to investigate a fuzzy algebraic

structure which is somehow independent of the crisp setting. For example, in classical

ring theory an ideal is a usp ideal if and only if its quotient is a usp ring. However, as

we shall prove in the example 7, this statement is not true for uspf ideals.

Section 4.1 provides the deﬁnition of uspf ideals and results about them. We prove that

the inverse image of Zadeh’s extension of uspf ideal is an uspf ideal which are constant

on Ker(f)(Proposition 32). On the other hand, the direct image of a uspf ideal of

Zadeh’s extension is not a uspf ideal (Example 8). Also, it is shown that all uspf ideals

are prime fuzzy ideals in accordance with the new deﬁnition of prime fuzzy ideal given

in Deﬁnition 15. It is shown how we can build a uspf ideal based on usp crisp ideal and

section 4.2 has new results on uspf ideals and contains questions and conjectures about

it. Finally, section 4.3 introduces the uniform strong radical in fuzzy settiong.

4.1 Introduction

Deﬁnition 27.[23] Let Rbe an associative ring with unity. A non-constant fuzzy ideal

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 24

I:R−→ [0,1] is said to be uspf ideal if there exists a ﬁnite subset Fsuch that

VI(xF y) = I(x)∨I(y), for any x, y ∈R. The set Fis called insulator of I.

Proposition 31.Iis uspf ideal of Riﬀ Iαis usp ideal of Rfor all I(1) < α ≤I(0).

Proof. Suppose Ia USPf ideal and let F⊆Rbe a ﬁnite set given by deﬁnition 27. Let

x, y ∈Rand I(1) < α ≤I(0) such that xF y ⊆Iα. Hence, I(x)∨I(y) = VI(xF y )≥α,

and thus I(x)≥αor I(y)≥α. Therefore, x∈Iαor y∈Iα. On the other hand,

suppose Iαis a usp ideal of Rfor all I(1) < α ≤I(0). According to Proposition 8each

Iαhas a ﬁnite set Fαsuch that if xFαy⊆Iαimplies x∈Iαor y∈Iα. Let a ﬁnite set

F=\

I(1)<α≤I(0)

Fα. Suppose VI(xF y )> I(x)∨I(y)and t=VI(xF y)for any x, y ∈R.

Note that t>I(x)∨I(y)and t≤I(xfy)for all f∈F. Hence, x, y 6∈ It, but xF y ⊆It

and thus (by hypothesis) x∈Itor y∈It, where we have a contradiction. Therefore,

VI(xF y) = I(x)∨I(y).

Corollary 18.If Iis a uspf ideal of a ring R, then R/Iαis a usp ring for all I(1) < α ≤

I(0).

Proof. It stems from the deﬁnition of usp ideal and the last proposition.

Corollary 19.If Iis uspf ideal, then Iis prime fuzzy ideal.

Proof. Since Iis uspf ideal, there exists a ﬁnite set F, where VI(xF y ) = I(x)∨I(y),

for any x, y ∈R. Note that xF y ⊆xRy. Hence, VI(xF y)≥VI(xRy). Therefore,

VI(xF y) = I(x)∨I(y)≥VI(xRy).

Proposition 32.If f:R−→ Sis a epimorphism of rings and Jis a uspf ideal of S, then

f−1(J)is a uspf ideal of Rwhich is constant on Ker(f).

Proof. As Jis uspf ideal, then there exists a ﬁnite set FJ(according to deﬁnition of uspf

ideal) and f−1(J)is a fuzzy ideal which is constant on Ker(f)by Proposition 11. Let

F=f−1(FJ), hence

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 25

Vf−1(J)(xF y) = VJ(f(xF y ))

=VJ(f(x)f(F)f(y))

=VJ(f(x)f(f−1(FJ))f(y))

=VJ(f(x)FJf(y))

=J(f(x)) ∨J(f(y))

=f−1(J)(x)∨f−1(J)(y).

Thus, f−1(FJ)is the insulator of a fuzzy ideal f−1(J).

Proposition 33.If Iis a uspf ideal of a ring R, then R/I is a usp ring.

Proof. As Iis uspf, there exists a ﬁnite set Fsuch that VI(xF y) = I(x)∨I(y), for

any x, y ∈R. Let F0= Ψ(F), where Ψis the natural homomorphism from Rto

R/I. Given x+I , y +I6=¯

0(i.e I(x), I(y)6=I(0)) in R/I. As Iis uspf, then we have

VI(xF y) = I(x)∨I(y). Hence, there exists f∈Fsuch that I(xf y) = I(x)∨I(y)6=I(0).

Therefore, xf y +I6=¯

0, where f+I∈F0and according to Lemma 2R/I is usp ring.

For the next result, consider I∗=II(0) ={x∈R:I(x) = I(0)}.

Proposition 34.If Iis a uspf ideal of a ring R, then R/I∗∼

=R/I.

Proof. Consider f:R−→ R/I, where f(x) = x+I. Note that r+I= 0 iﬀ I(r) = I(0).

Thus, Ker(f) = I∗and by the isomorphism theorem [36] we have R/I∗∼

=R/I.

Corollary 20.If f:R−→ Sis an epimorphism and Iuspf ideal of Rwhich is constant

on Ker(f), then R/I ∼

=S/f(I).

Proof. Deﬁne h:R−→ S/f(I)∗and h(x) = f(x) + f(I)∗. Thus, his onto and

Ker(h) = I∗. Applying the isomorphism theorem, R/I∗∼

=S/f(I)∗. Thus, R/I ∼

R/I∗∼

=S/f(I)∗∼

=S/f(I);

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 26

The next proposition shows us how it is possible to build a uspf ideal based on usp crisp

ideal.

Proposition 35. Let Jbe an ideal (crisp) of R. Deﬁne I:R−→ [0,1] as

I(x) = 









1, if x = 0;

α, if x ∈J\ {0};

0, if x /∈J,

where 0< α < 1. Then:

i) Iis a fuzzy ideal;

ii) Iis uspf ideal iﬀ Jis usp ideal.

Proof. i) Note that all α-cuts of Iare I∗= (0),Iα=Jand I0=R, according to

Proposition 10 Iis a fuzzy ideal of R. ii) Suppose Iis uspf ideal. We will prove that

Jis an usp ideal according to Proposition 9. Thus, let x, y /∈J, as Iis uspf, there

is a ﬁnite set F, where VI(xF y) = I(x)∨I(y)=0. Since Fis ﬁnite, there exits

f∈Fwhere I(xfy)=0, then xf y /∈J. On the other hand, suppose Jis usp ideal

of R, hence there exists a ﬁnite set Ffor Jaccording to deﬁnition of usp crisp ideal.

Thus, given x, y ∈Rwe have the following cases: 1) If x, y = 0, then we have triviality

VI(xF y) = I(0) = I(x)∨I(y) = I(0); 2) If x∈Jor y∈J, then xF y ⊆J. Thus,

VI(xF y) = α=I(x)∨I(y); 3) If x /∈Jand y /∈J, then there exists f∈Jsuch that

xfy /∈J. Thus, VI(xF y) = 0 = I(x)∨I(y).

Corollary 21.Let Ibe a non-constant fuzzy ideal of Rand deﬁne:

M(x) = 









I(0), if x = 0;

α, if x ∈I∗\ {0};

I(1), if x /∈I∗.

Then, Mis uspf ideal of Riﬀ I∗is usp ideal of R.

Proof. Straighforward

Corollary 22.Let I1⊂I2⊂ ··· ⊂ In=Rbe any chain of usp ideals of a ring R. Let

t1, t2, . . . , tnbe some numbers in [0,1] such that t1> t2> . . . > tn. Then the fuzzy

subset Ideﬁned by

I(x) = (t1, if x ∈I1

ti, if x ∈Ii\Ii−1, i = 2, . . . , n,

is a uspf ideal of R.

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 27

Example 7.Consider Zthe ring of integers and 4Z={x∈Z:x= 4q, q ∈Z}. Deﬁne

a fuzzy set as: I(x) = 









1, if x = 0;

1/2, if x ∈4Z\ {0};

0, if x /∈4Z.

Iis a fuzzy ideal, since its all α-cuts (I1= (0), I1/2= 4Z, I0=Z) are ideals. Moreover,

Iis not uspf ideal, since 4Zis not prime ideal, acccording to Proposition 35. Note

that I∗= (0) is usp ideal. Hence, R/I∗is a usp ring. Applying the Proposition 34

R/I ∼

=R/I∗. Therefore, R/I is a usp ring, but Iis not uspf ideal.

Example 8.Let f:Z−→ Z4be deﬁned by f(x)=[x]4=x mod 4. The function fis

an epimorphism with kernel 4Z. Consider

I(x) = 









1, if x = 0;

1/2, if x ∈3Z\ {0};

0, if x /∈3Z.

and then

f(I)(y) = (1, if x = 0;

1/2, if x 6= 0.

Clearly Iis uspf ideal of Z, but f(I)is not uspf ideal of Z4, since I1/2=Z4is not usp

ideal.

4.2 Extra Results on Uspf Ideals

This section ampliﬁes results about uspf ideals. The ﬁrst one (Proposition 36) is geared

to commutative rings. However, it may be valid for noncommutatives (Conjecture 2).

The Proposition 8brings the diﬀerence between crisp and fuzzy setting by showing the

behavior of Zadeh’s extension on uspf ideals. The results in this section was published

in [37].

Proposition 36.If Iis a non-constant fuzzy ideal of a commutative ring R, then there

exists a uspf ideal Ksuch that I⊆K.

Proof. Consider the crisp ideal I∗={x∈R:I(x)> I(1)}. By Zorn’s Lemma, there

exists a maximal ideal Mof Rcontaining I∗. Now we can deﬁne the following fuzzy set:

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 28

K(x) = (I(0) if x ∈M,

I(1) otherwise.

Clearly, Kis a fuzzy ideal and I⊆K. Now, consider the ﬁnite set F={1}. Thus,

VK(xF y) = K(xy)for any x, y ∈R.

If x∈Mor y∈M, then xy ∈Mand then K(xy) = I(0) = K(x)∨K(y). On the other

hand, as Ris commutative, Mis completely prime, hence if x /∈Mand y /∈M, then

xy /∈M. Therefore, K(xy) = I(1) = I(1) ∨I(1) = K(x)∨K(y).

Conjecture 1.According to the deﬁnition of fuzzy maximal ideal given in Deﬁnition 17,

the set Kin the demonstration of Proposition 36 is a fuzzy maximal ideal.

Conjecture 2.The Propostion 36 can be extended to noncommutative rings.

Proposition 37.Let f:R−→ Sis a epimorphism of commutative and non usp rings. If

Iis a uspf ideal of Rwhich is constant on Ker(f), then f(I)is not a uspf ideal of S.

Proof. As Iis constant on Ker(f), then by Proposition 34 and Corollary 20 we have:

R/I∗∼

=R/I ∼

=R/f(I)∼

=R/f(I)∗. As I∗is usp ideal, then R/I∗is usp ring. Hence,

R/f(I)∗is usp ring. Thus, f(I)∗is usp ideal. As we know f(I)∗⊆f(I)αfor all α∈[0,1].

But Sis commutative and f(I)∗is Prime, hence f(I)∗is maximal, this last stantement

implies f(I)α=Sfor all α6=I(0) and by hypoteses Sis not usp ring. Therefore, f(I)

is not uspf.

Question 2.The Proposition 37 shows us that uspf ideals cannot be preserved by Zadeh’s

extension. Thus, we ask: Under which conditions can Zadeh’s extension preserve the uspf

ideals? This question still open.

Proposition 38.If Iand Pare fuzzy ideals of a ring Rwith Puspf ideal, then I∩Pis

uspf ideal of R.

Proof. Note that: V(I∩P)(xF y ) = (VI(xF y)) ∧(VP(xF y)) = (VI(xF y)) ∧(P(x)) ∨

P(y)≤P(x)∨P(y).

Proposition 39.Any uspf ideal contains properly another uspf ideal

Proof. Suppose Iuspf ideal of a ring R. Let P=1

2I⊂Ideﬁned by P(x) = 1

2I(x)for

all x∈R. Hence, VP(xF y ) = ^I(xF y)

2=I(x)

2∨I(y)

2=P(x)∨P(y).

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 29

The next proposition tell us about the following question: If a fuzzy ideal has at least

one usp α-cut, what can we say about this ideal. Is it a uspf ideal?

Proposition 40.Let Ia nonconstant fuzzy Ideal of a Integral Domain Rand Ris not a

usp ring and Itis usp ideal for some I(1) < t ≤I(0). If k6=tand Ik6=It, then Ikis not

usp ideal. Hence, Iis not a uspf ideal.

Proof. When Ik=Ris trivial. Now suppose Ik6=Rand note that in a Integral Domain

if Iis usp ideal, then Iis a Maximal ideal. Thus, It⊂Ikis impossible, since Itis

Maximal. If Ik⊂Itimplies Iknot maximal, then Iknot a usp ideal.

4.3 US Fuzzy Radical

Since its inception, general theory of radicals has proved to be fundamental for the

structure of ring theory. A better understanding of radical of a fuzzy ideal can give to

us some information about its nature. The crisp Uniformly Strongly Prime radical (US

radical) of a ring Rwas deﬁned to be the intersection of all usp ideals of R. Olson

[3] located this radical in the lattice of radical classes and proved that US radical is

independent of Jacobson and Brown-Mccoy radical. In this section we deﬁned the US

radical of a fuzzy ideal in the standard way by comparing with two new notions.

Deﬁnition 28.[3] The US radical of a crisp ideal Iis US(I) = ∩{P⊆R:P⊇

I and P is usp ideal of R}.

Deﬁnition 29.[23] Let Ibe a fuzzy ideal of R.The uniformly strongly fuzzy radical of I

is u

√I=∩{P⊆R:P⊇I and P is uspf}.

The radical uspfR of a ring Ris deﬁned as u

√0. Clearly

√R=∩{P⊆R:P is uspf ideal of R}.

Remark 1.According to [23] the quotient R/ u

√Ris usp ring and R/(u

√R)∗∼

=R/ u

√R.

Clearly, u

√Iis an ideal, and if Iis a unif. strongly prime fuzzy ideal, then u

√I=I.

Proposition 41.If Iis a fuzzy ideal of a ring R, then u

√Iis a uspf ideal of R.

Proof. Consider SI={P⊆R:P⊇I and P is uspf}and F=\

P∈SI

FP, where

FPis a ﬁnite set (insulator) of P. Clearly Fis a ﬁnite set. Given x, y ∈R, hence,

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 30

√I(xF y) = ^

\

P∈SI

P(xF y)

=^

^

P∈SI

P(xF y)

=^

P∈SI^P(xF y)=

P∈SI

(P(x)∨P(y)) = ^

P∈SI

P(x)∨^

P∈SI

P(y) = u

√I(x)∨u

√I(y).

Proposition 42.If I , J are a fuzzy ideals of a ring R, then:

(i) if I⊆J, then u

√I⊆u

√J;

(ii) u

√I=u

√I;

(iii) Iα⊆(u

√I)α;

(iv) If Iis unif. strongly prime fuzzy ideal, then u

√Iα= ( u

√I)α;

(v) u

√I∩J⊆u

√I∩u

√J.

Proof. (i) u

√J=\

P∈SJ

P⊇\

P∈SI

P=u

√I. (ii) It is easy to see that u

√I⊆u

√I. On the

other side, let’s show SI⊆ Su

√I. In fact, let P∈ SI, then P⊇Iusing (i) P=u

√P⊇u

√I.

(iii),(iv) and (v) are straightforward.

Proposition 43.Let f:R−→ Sbe a homomorphism of rings and Ia fuzzy ideal of R.

Then:

1)f(I)⊆f(u

√I)⊆u

qf(u

√I);

2) I⊆f−1(u

pf(I)).

Proof. 1) Straightforward.

2) As f(I)⊆u

pf(I), then f−1(f(I)) ⊆f−1(u

pf(I)). Thus, I⊆f−1(f(I)) ⊆f−1(u

pf(I)).

Proposition 44.Let f:R−→ Sbe a homomorphism of rings and Ia SP fuzzy ideal of

R. Then, f(u

√I)⊆u

pf(I).

Proof. As Iis SP fuzzy u

√I=I, then u

pf(I) = u

qf(u

√I). Thus, f(u

√I)⊆u

qf(u

√I) =

pf(I).

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 31

4.3.1 Extra Results on US Radical

Proposition 45.If Iis a fuzzy ideal of Rand P⊇Ia uspf ideal, then ∩Pt=U S(It)for

any t∈(I(1), I(0)], where Pt, Itare t-cuts of Pand Irespectively.

Proof. Straightforward.

Deﬁnition 30.The radical uspfR-inf of a fuzzy ideal Iis αI:R−→ [0,1], where

αI(x) = ∧{t:x∈US(It)}. The radical uspfR-sup is βI:R−→ [0,1], where βI(x) =

∨{t:x∈US(It)}.

Proposition 46.If Iis a fuzzy ideal of R, then u

√I≥βI≥αI.

Proof. Clearly, βI≥αI. For u

√I≥βIconsider x∈US(It). According to Proposition 45,

x∈ ∩Pt. Thus, ∩P(x)≥t, where I⊆Pand Pis uspf ideal. Therefore, u

√I≥βI.

Question 3.Under which conditions u

√I=βI?

Proposition 47.If Iis a non-constant fuzzy Ideal of a commutative ring R, then u

√I(0) =

I(0) and u

√I(1) = I(1).

Proof. According to Proposition 36 there exists a uspf ideal K⊇I. Thus, u

√I(0) ≤

K(0) = I(0) and u

√I(1) ≤K(1) = I(1). By the deﬁnition of uspfR radical u

√I⊇I, then

√I(0) ≥I(0) and u

√I(1) ≥I(1).

Conjecture 3.The Proposition 47 is valid in associative rings with unit.

4.4 The Fuzzy m- and t-systems

An m-system is a generalization of multiplicative systems. In the ring theory a set Mis

am-system if for any two elements x, y in Mthere exists rin Rsuch that the product

xry belongs M. It is not hard to see that an ideal is prime iﬀ its complement is a m-

system (see Mccoy [38]). On the other hand we have the t-systems which are sets where

if any two elements x, y in Mthere exists a ﬁnite set Fsuch that xfy belongs Mfor

some fin F. Clearly a t-system is a m-system. Olson [3] proved that Iis a uniformly

strongly prime ideal iﬀ its complement is a t-system. Therefore, we have a tool to deal

with primeness and uniform strong primeness.

In this chapter we will introduce the m-system in a fuzzy setting based on the deﬁnition

of prime fuzzy ideals without α-cut dependence, given by Navarro [13] in 2012. The

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 32

t-system is also introduced, and we have another characterization of uspf ideal. At the

end of the chapter a method to count the number of the uspf ideals in a ﬁnite ring is

introduced.

Deﬁnition 31.[38] A subset Kof a ring Ris called a m-system if for any two elements

x, y ∈Kthere exists r∈Rsuch that xry ∈K.

Deﬁnition 32.[3] A subset Mof a ring Ris called a t-system if there exists a ﬁnite set

F⊆Rsuch that for any two elements x, y ∈Mthere exists f∈Fsuch that xfy ∈M.

In the last Deﬁnition Fwill be called the insulator of M. The empty set will be a

t-system by deﬁnition.

Proposition 48.[38] If Mis a t-system, then Mis a m-system.

Proposition 49.[38]Iis a prime ideal of a ring Riﬀ R\I(the complement of Iin R)

is a m-system.

Proposition 50.[3] An ideal Iis usp of a ring Riﬀ R\I(the complement of Iin R) is

at-system.

Proposition 51.[3] If Iand Pare ideals of a ring Rwith Pusp ideal, then I∩Pis usp

ideal.

4.4.1 The Fuzzy m- and t-Systems

For the next deﬁnition consider xRy ={xry :r∈R}.

Deﬁnition 33.Let Rbe an associative ring with unity. A non-constant fuzzy set K:

R−→ [0,1] is said to be fuzzy m-system if WK(xRy) = K(x)∧K(y), for any x, y ∈R.

Proposition 52.If Kis fuzzy subset of a ring Rsuch that Kαis a m-system for all α-cuts,

then WK(xRy)≥K(x)∧K(y),

Proof. Let x, y ∈Rand t=K(x)∧K(y). As Ktis m-system and x, y ∈Ktthen there

exists r∈Rsuch that xry ∈Kti.e K(xry)≥t. Hence, WK(xRy)≥t.

Question 4.Under which conditions can we have the following result: is Ka fuzzy

m-system of Riﬀ Kαis an m-system for all α-cuts?

For the next results consider Pthe fuzzy ideal and Pc= 1 −Pthe complement of Pin

Corollary 23.Pis a prime fuzzy ideal of Riﬀ Pc(the complement of Pin R) is a fuzzy

m-system.

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 33

Proof. Suppose Pprime fuzzy, then VP(xRy) = P(x)∨P(y)for any x, y ∈R. Hence,

WPc(xRy) = W(1 −P(xRy)) = 1 −VP(xRy)=1−(P(x)∨P(y)) = (1 −P(x)) ∧

(1 −P(y)) = Ic(x)∧Ic(y). On the other side, suppose Pca fuzzy m-system, then

WPc(xRy) = Pc(x)∧Pc(y)for any x, y ∈R. Thus, 1−VP(xRy)=1−(P(x)∨P(y)).

Therefore VP(xRy) = P(x)∨P(y).

For the next deﬁnition consider the subset xF y ={xf y :f∈F}of ring R.

Deﬁnition 34.Let Rbe an associative ring with unity. A non-constant fuzzy set M:

R−→ [0,1] is said to be fuzzy t-system if there exists a ﬁnite subset Fsuch that

WM(xF y) = M(x)∧M(y), for any x, y ∈R.

Proposition 53.Iis a uspf ideal of Riﬀ Ic(the complement of Iin R) is a fuzzy t-system.

Proof. Suppose Iuspf, then there exists a ﬁnite set Fwhere VI(xF y ) = I(x)∨I(y)for

any x, y ∈R. Hence, WIc(xF y) = W(1 −I(xF y )) = 1 −VI(xF y )=1−(I(x)∨I(y)) =

(1 −I(x)) ∧(1 −I(y)) = Ic(x)∧Ic(y). On the other side, suppose Ica fuzzy t-system,

then there exists a ﬁnite set Fwhere WIc(xF y) = Ic(x)∧Ic(y)for any x, y ∈R. Thus,

1−VI(xF y)=1−(I(x)∨I(y)). Therefore VI(xF y) = I(x)∨I(y).

Proposition 54.If Mis a fuzzy t-system of R, then Mαis a t-system for all α-cuts.

Proof. As Mis a fuzzy t-system there exists a ﬁnite set F, where VI(xF y ) = I(x)∨I(y)

for any x, y ∈R. Let x, y ∈Mα, then WM(xF y) = M(x)∧M(y)≥α. Since Fis a

ﬁnite set, there exists f∈Fsuch that M(xfy)≥α. Thus, xfy ∈Mα. Therefore, Mα

is a t-system.

Question 5.Under which conditions can we have the following result: if Kis a fuzzy

t-system of R, then is Kan m-system?

The next example help us to count (in certain way) the number of uspf ideals of a ﬁnite

ring R. We recommend the reading of Chapter 5before.

Example 9.Let I, J fuzzy subsets of a ring R. Deﬁne the following relation: I∼Jiﬀ

I, J induce the same α-cuts. Clearly, ∼is an equivalence relation.

Consider R=Z12 the integers mod 12. According to the diagram below we can count

the possible chains of crisp ideals ending with Z12. Hence, we have 15 following chains:

Chapter 4. Uniformly Strongly Prime Fuzzy Ideals 34

{0}

Z12

Figure 4.1: Ideals of Z12

1. {0} ⊂ Z12

2. {0} ⊂ 6Z⊂Z12

3. {0} ⊂ 6Z⊂3Z⊂Z12

4. {0} ⊂ 3Z⊂Z12

5. {0} ⊂ 6Z⊂2Z⊂Z12

6. {0} ⊂ 2Z⊂Z12

7. {0} ⊂ 4Z⊂Z12

8. {0} ⊂ 4Z⊂2Z⊂Z12

9. 2Z⊂Z12

10. 3Z⊂Z12

11. 3Z⊂6Z⊂Z12

12. 4Z⊂2Z⊂Z12

13. 4Z⊂Z12

14. 6Z⊂Z12

15. 6Z⊂2Z⊂Z12

According to Propositon 75 if Iis a uspf ideal of Z12, then the α-cuts are usp ideals

for all I(1) < α ≤I(0). Hence, we can count the numbers of uspf ideals of Z12. Since

Z12 has only two usp ideals (2Z, 3Z), then it has only 5 uspf ideals in Z12 under the

equivalence relation ∼.

Chapter 5

Some Properties of Fuzzy Ideals

In this chapter it is investigated the homomorphic image of fuzzy subring, fuzzy ideal,

fuzzy prime, and fuzzy irreducible ideals of a ring by Zadeh’s extension and how it may

inﬂuence in the homomorphism of the fuzzy ideal lattices. Finite-valued fuzzy ideals and

their relations with Artinian (Noetherian) rings are also described.

Section 5.1 provides an overview on the theory of fuzzy rings. Section 5.2 contains

the demonstration of the following result: if Gand Hare isomorphic rings, then the

respective family of ideals are isomorphic lattices. Section 5.3 contains the investigation

of some results about fuzzy ring theory similarly to [39]. It is proved that F IFV (R),

which is the set of ﬁnite-valued fuzzy ideals of R, is a sublattice of LF I (R)(the lattice of

all fuzzy ideals of R). Moreover, a condition is shown in order to a fuzzy set to belong to

F IF V (R)(based in a chain of ideals of R)i.e. the fuzzy set is a ﬁnite-valued fuzzy ideal

iﬀ there exists a certain kind of chain of ideals. This result entails that, if Ris Artinian

ring with unity, then a fuzzy ideal can be written in terms of the chain of ideals. At the

end of the section, it is proved that LI(R),i.e. the lattice of ideals of R, is isomorphic

to a sublattice of F IF V (R).

5.1 Preliminaries

This section explains some deﬁnitions and results that will be required in the next sec-

tions.

A fuzzy set µis ﬁnite-valued, whenever Im(µ)is a ﬁnite set.

The set of all fuzzy subrings and ideals of the ring Rare denoted by LF(R)and LF I (R),

respectively.

Chapter 5. Some Properties of Fuzzy Ideals 36

Deﬁnition 35.Let µ, ν be any two fuzzy subsets of ring Rthe product µν is deﬁned as

follows:

(µν)(x) =











x=y·z

(µ(y)∧ν(z)), where y, z ∈R

0,if xis not expressible as x=y·z

for all y, z ∈R.

Deﬁnition 36.A non-constant fuzzy ideal µof a ring Ris called fuzzy prime if for any

fuzzy ideals µ1and µ2of Rthe condition µ1µ2⊆µimplies that either µ1⊆µor µ2⊆µ.

According to Navarro [13] if Iis a fuzzy prime ideal, then Iis a fuzzy prime. Then, all

results proved for fuzzy prime can be used for fuzzy prime ideal.

Deﬁnition 37.A fuzzy ideal µof a ring Ris called fuzzy irreducible if it is not a ﬁnite

intersection of two fuzzy ideals of Rproperly containing µ: otherwise µis termed fuzzy

reducible.

Some properties of fuzzy rings/ideals can be veriﬁed in the works [31,40–42]. Note that

for any fuzzy subring/fuzzy ideal µof a ring R, if for some x, y ∈R,µ(x)< µ(y), then

µ(x−y) = µ(x) = µ(y−x).

Theorem 24.[31] If µis any fuzzy subring/fuzzy ideal of a ring R, then each level subset

µt={x∈R:µ(x)≥t}where 0≤t≤µ(0) is a subring/an ideal crisp of R. In

particular, if Rhas unity, Im(µ)⊆[µ(1), µ(0)].

Theorem 25.[31] A fuzzy subset µof a ring Ris a fuzzy subring/fuzzy ideal of Riﬀ the

level subsets µt,(t∈Im µ)are subrings/ideals of R.

In general, the product of two fuzzy ideals may not be a fuzzy ideal.

Proposition 55.The family of fuzzy subrings/fuzzy ideals of a ring Ris closed under

intersection.

Proposition 56.Let µbe any fuzzy subring and νany fuzzy ideal of a ring R. Then µ∩ν

is a fuzzy ideal of the crisp subring {x∈R:µ(x) = µ(0)}.

Proposition 57.Let I1⊂I2⊂ ··· ⊂ In=Rbe any chain of ideals of a ring R. Let

t1, t2, . . . , tnbe some numbers in the interval [0,1] such that t1> t2>··· > tn. Then

the fuzzy subset µof Rdeﬁned by

Chapter 5. Some Properties of Fuzzy Ideals 37

µ(x) = (t1, if x ∈I1

ti, if x ∈Ii\Ii−1, i = 2, . . . , n,

is a fuzzy ideal of R.

Theorem 26 ([40]).Let Rbe a ring with unity. Ris Artinian iﬀ every fuzzy ideal of R

is ﬁnite-valued.

Theorem 27 ([41]).If a Γ-ring Mis Artinian, then every fuzzy ideal of Mis ﬁnite-valued.

5.2 The Isomorphism

In this section it is proved that for a given pair of rings R, S the lattices of fuzzy ideals

of Rand Sare isomorphic whenever Rand Sare isomorphic rings.

Proposition 58.Zadeh’s extension preserves fuzzy subrings. Let f:R−→ Sbe a

homomorphism of rings and µa fuzzy subring of the ring R. Then f(µ)is a fuzzy

subring of the ring S.

Proof. Let x, y ∈S. Suppose either x /∈f(R)or y /∈f(R). Then according deﬁnition

14 f(µ)(x) = 0 or f(µ)(y) = 0. Thus f(µ)(x)∧f(µ)(y) = 0 ≤f(µ)(x−y)and

f(µ)(x)∧f(µ)(y) = 0 ≤f(µ)(xy).

Now suppose x, y ∈f(R). Observe that fis a homomorphism, f(R)is a subring and

x−y, xy ∈f(R). Moreover, for demonstration below we will use the following equality

∨{A∪B}= (∨{A})∨(∨{B}).

f(µ)(x−y) = ∨{µ(z) : f(z) = x−y} ≥ ∨{µ(m−n) : f(m) = xand f(n) = y} ≥

∨{µ(m)∧µ(n) : f(m) = xand f(n) = y}=∨{∪{µ(m)∧µ(n) : f(m) = x}:f(n) =

y}=∨{∨{µ(m)∧µ(n) : f(m) = x}:f(n) = y}=∨{∨{µ(m) : f(m) = x} ∧

µ(n) : f(n) = y}= (∨{µ(m) : f(m) = x})∧(∨{µ(n) : f(n) = y}) = f(µ)(x)∧f(µ)(y).

f(µ)(xy) = ∨{µ(z) : f(z) = xy} ≥ ∨{µ(mn) : f(m) = xand f(n) = y} ≥ ∨{µ(m)∧

µ(n) : f(m) = xand f(n) = y}=∨{∪{µ(m)∧µ(n) : f(m) = x}:f(n) = y}=

∨{∨{µ(m)∧µ(n) : f(m) = x}:f(n) = y}=∨{∨{µ(m) : f(m) = x} ∧ µ(n) : f(n) =

y}= (∨{µ(m) : f(m) = x})∧(∨{µ(n) : f(n) = y}) = f(µ)(x)∧f(µ)(y).

Proposition 59 ([43]).Epimorphism preserves fuzzy ideals. Let f:R−→ Sbe an

epimorphism of rings and µa fuzzy ideal of the ring R. Then f(µ)is a fuzzy ideal of the

ring S.

Chapter 5. Some Properties of Fuzzy Ideals 38

Proof. Let x, y ∈S=f(R), then x−y, xy ∈f(R).

f(µ)(x−y)≥f(µ)(x)∧f(µ)(y)is similar to Proposition 58.

f(µ)(xy) = ∨{µ(z) : f(z) = xy} ≥ ∨{µ(mn) : f(m) = xand f(n) = y} ≥ ∨{µ(m)∨

µ(n) : f(m) = xand f(n) = y}= (∨{µ(m) : f(m) = x})∨(∨{µ(n) : f(m) = y}) =

f(µ)(x)∨f(µ)(y).

Since for any homomorphism of rings f:A−→ B,f(A)is also a ring, and f:A−→

f(A)is an epimorphism, then it is reasonable to say that homomorphism induces the

preservation of fuzzy ideals.

Deﬁnition 38.For any two fuzzy ideals µand νof R, deﬁne µ∧ν=µ∩νand µ∨ν=

∩{η:ηis a fuzzy ideal of Rsuch that η≥µ, ν}.

Proposition 60.Let Rbe a ring. Then LF I (R)is a complete lattice under ∧and ∨.

Proof. See [44].

As it is known, homomorphism preserves algebraic structures. As we will prove, the

theorem below shows that Zadeh’s extension preserves certain algebraic properties.

Theorem 28.Let R, S be rings. If R∼

=S, then the lattices LF I (R)and LF I (S)are

isomorphic.

Proof. As fis an isomorphism, then f(µ)(y) = µ(x)for y=f(x)(1). Let µ, ν ∈LF I (R)

and y∈R. Then:

f(µ∧ν)(y) = µ∧ν(x) (1) = µ(x)∧ν(x) = f(µ)(y)∧f(ν)(y)

f(µ)∨f(ν)(y) = ∧{η:η≥f(µ), f (ν)}(y) = ∩{η(y) : η(y)≥f(µ)(y), f (ν)(y)}=

∩{η(f(x)) : η(f(x)) ≥µ(x), ν(x))}=∩{η◦f(x) : η◦f(x)≥µ(x), ν(x))}=

∩{η0(x) : η0(x)≥µ(x), ν(x))}η0=η◦f}=µ∨ν(x) = f(µ∨ν)(y) (1)

To prove that Zadeh’s extension is bijective, let f(µ) = f(ν)then f(µ)(y) = f(ν)(y)

for all y∈S. By deﬁnition ∨{µ(x) : f(x) = y}=∨{ν(x) : f(x) = y}. By (1)

µ(x) = ν(x). Therefore, µ(x) = ν(x)for all x∈Rand then µ=ν. On the other

hand, let µ∈LF I (S)and deﬁne νsuch that ν(x) = µ(y)where y=f(x). Thus

f(ν)(y) = ∨{ν(x) : f(x) = y}=ν(x) = µ(y)for all y∈S. Therefore f(ν) = µ.

The converse of this theorem is not true as shown by the following example:

Chapter 5. Some Properties of Fuzzy Ideals 39

Example 10.Consider the rings Qand Rof rational and real numbers, respectively. Both

contain only two ideals: (0) and their own ring. Then |LF I (Q)|=|LF I (R)|and in this

case it is not diﬃcult to show that the lattices LF I (Q),LF I (R)are isomorphic.

The next propositions are corollaries of the Theorem 28.

Proposition 61.Let f:R−→ Sbe an isomorphism of rings. If µis a prime fuzzy of R,

then f(µ)is a fuzzy prime of S.

Proof. Let ν1, ν2be fuzzy ideals of Ssuch that ν1ν2⊆f(µ), then by using the Theorem 28

there exist µ1, µ2fuzzy ideals such that ν1=f(µ1)and ν2=f(µ2). Since ν1ν2⊆f(µ),

then µ1µ2⊆µ. In accordance with the hypothesis, µ1⊆µor µ2⊆µ. Therefore

ν1⊆f(µ)or ν2⊆f(µ).

Proposition 62.Let f:R−→ Sbe an isomorphism of rings. If µis a fuzzy irreducible

of R, then f(µ)is a fuzzy irreducible of S.

Proof. We will show the contrapositive. Suppose f(µ)is a fuzzy reducible of S, then

there exist ν1, ν2such that f(µ) = ν1∩ν2. By the Theorem 28,ν1∩ν2=f(µ1)∩f(µ2) =

f(µ1∩µ2). Thus µ=µ1∩µ2. Therefore µis reducible.

5.3 An Equivalence of Fuzzy Ideals

In this section we look at ﬁnite-valued fuzzy subsets. We prove that F IF V (R)is a

sublattice of LF I (R), and we give a condition for the ideal to belong to F IF V (R)based

on chain of ideals of R. A consequence of this fact is that if Ris an Artinian ring with

unity, then a fuzzy ideal can be written in terms of the chain of ideals of R. At the end

of the section, we prove that LI(R),i.e. the lattice of ideals of R, is isomorphic to a

sublattice of F IF V (R).

Theorem 29.F IF V (R)is a sublattice of LF I (R).

Proof. If µ, ν is a ﬁnite-valued fuzzy ideal, clearly µ∧νand µ∨νare ﬁnite-valued fuzzy

ideal.

Theorem 30.Let µbe a fuzzy subset of Rsuch that Im(µ) = {t1, t2, . . . , tn}where

t1> t2>··· > tn. Hence µcan be written as

Chapter 5. Some Properties of Fuzzy Ideals 40

µ(x) = 









t1, if x ∈B1

t2, if x ∈B2

tn, if x ∈Bn,

with Bi∩Bj=∅for i6=jand

[

i=1

Bi=R.

Proof. Let B1={x∈R:µ(x) = t1},Bk={x∈R\Sk−1

j=1 Bj:µ(x) = tk},k∈

{2,··· , n}.

Observation 1.Given the sets Biintroduced in the proof of theorem 30, let Im=

[

i=1

Bi,

then Im={x∈R:µ(x)≥tm},Bi+1 =Ii+1\Iiand In=

[

i=1

Bi=R. Thus it can be

written

µ(x) = 









t1, if x ∈I1

t2, if x ∈I2\I1

tn, if x ∈In\In−1.

Where Ii,i∈ {1, . . . , n}are ideals of ring Rby Theorem 24.

The next Corollary 31 gives a condition to µbe a ﬁnite-valued fuzzy ideal.

Corollary 31.µis a ﬁnite-valued fuzzy ideal, if and only if, there exists a chain of ideals

I1⊂I2⊂ ··· ⊂ In=Rsuch that Ii={x∈R:µ(x)≥ti},i∈ {1, . . . , n}and

µ(x) = 









t1, if x ∈I1

t2, if x ∈I2\I1

tn, if x ∈In\In−1.

Proof. (⇒) If µis a ﬁnite-valued fuzzy ideal, then it is possible to build the sequence

like observation 1.

(⇐) Theorem 57.

Chapter 5. Some Properties of Fuzzy Ideals 41

In particular, if Ris a ﬁnite ring, then all fuzzy ideals are completely determined by the

chains of ideals in R.

Corollary 32.Let Rbe an Artinian ring with unity. µis a fuzzy ideal of R, if and only if,

there exists a ﬁnite chain I1⊂I2·· · ⊂ Insuch that µsuch that Ii={x∈R:µ(x)≥ti},

i∈ {1, . . . , n}and

µ(x) = 









t1, if x ∈I1,

t2, if x ∈I2\I1

tn, if x ∈In\In−1.

Proof. Immediately from Theorem 26 and Corollary 31.

Corollary 33.Let Mbe an Artinian Γ-ring M .µis a fuzzy ideal of R, if and only

if, there exists a ﬁnite chain I1⊂I2··· ⊂ Insuch that Ii={x∈R:µ(x)≥ti},

i∈ {1, . . . , n}and

µ(x) = 









t1, if x ∈I1,

t2, if x ∈I2\I1

tn, if x ∈In\In−1.

Proof. Immediately from Theorem 27 and Corollary 31.

In F IFV(R)it is deﬁned the following relation: let µ, ν ∈F IFV(R)then:

1) µ∼νiﬀ µ, ν induce the same ideal chains.

2) µ≡νiﬀ Iµ

1=Iν

Clearly ∼and ≡are equivalence relations. Moreover if µ∼νthen µ≡ν.

Example 11.Consider the ring of integers Zand deﬁne:

µ(x) = (1, if x ∈2Z

1/2, if x ∈Z\2Z

η(x) = (1/3, if x ∈2Z

1/4, if x ∈Z\2Z

Chapter 5. Some Properties of Fuzzy Ideals 42

ν(x) = (1/3, if x ∈4Z

1/4, if x ∈Z\4Z

β(x) = 









1, if x ∈4Z

1/2, if x ∈2Z\4Z

1/4, if x ∈Z\2Z

µ∼η, µ ν, µ ≡η, ν ≡β.

As it has been shown before, any ﬁnite-valued fuzzy ideal of Rdetermines a chain of

ideals in Rof type I1⊂I2⊂ ··· ⊂ In=Rsuch that Ii={x∈R:µ(x)≥ti},

i∈ {1, . . . , n}. Now it is possible to count the number of fuzzy ideals of a ﬁnite ring

based on ∼or ≡since in a ﬁnite ring it is possible to count the number of ideal chains.

Example 12.Consider Z5the integers mod 5. Herein there are only 2 chains, i.e. {0} ⊂

Z5and Z5. Hence under ∼there are only 2 fuzzy ideals.

Although this work provides two equivalence relations in F IFV(R)namely ∼and ≡,

in what follows, only ≡is investigated. Consider LI(R)which is the lattice of all crisp

ideals of R. The equivalence classes modulo ≡are:

L={µI:I∈LI(R)}, where

µI:R−→ [0,1],µI(x) = (1if x ∈I,

0if x ∈R\I, for all I∈LI(R).

Proposition 63.The set L={µI:I∈LI(R)}is a sublattice of F IF V (R).

Proof. It is easy to see that for any two ideals Iand Jof R,µI∧µJ=µI∩Jand

µI∨µJ=µI∪J.

Proposition 64.The map f:LI(R)−→ LF I (R)deﬁned by f(I) = µI, for all I∈LI(R),

is a lattice embedding and LI(R), L are isomorphic lattices.

Proof. It is enough to see that: if I, J ∈LI(R), then

f(I∩J) = µI∩J=µI∧µJ=f(I)∧f(J).

f(I∪J) = µI∪J=µI∨µJ=f(I)∨f(J)

Chapter 5. Some Properties of Fuzzy Ideals 43

5.4 Final Remarks

The theorem 28 seems very simple, but it brings relevant information to the study of

fuzzy algebra, because it tells us that we can look at the lattice of fuzzy ideals in the

same manner that we look at the lattice of crisp ideals. Finally, it worth to think the

theorem 28 for semisimple rings based on pure deﬁnition of prime fuzzy ideal.

Chapter 6

Prime Ideals and Fuzzy Prime Ideals

Over Noncommutative Quantales

In this chapter we propose a new concept of prime ideals in noncommutative quantales.

The usual deﬁnition of prime ideal is preserved as a completely prime ideal. In this

investigation it is proved that these two concepts coincide in commutative quantales,

but are no longer valid in the noncommutative setting. Also, the notions of strong

and uniform strong primeness as well as the fuzzy version of prime ideal and uniformly

strongly prime ideal are introduced in quantales. All these studies in this chapter were

submited to the fuzzy sets and systems journal.

6.1 Introduction

In 2013, Lingyun Yang and Luoshan Xu [45] deﬁned a prime ideal in quantales based on

elements of quantale. After that they built the rough prime ideal in quantales over this

concept. In 2014, Qingjun Luo and Guojun Wang [46] used the same deﬁnition of prime

ideals of quantales to write an investigation called roughness and fuzziness where the ﬁrst

ideas on semi-prime, primary and strong primeness are presented. As it is known, ideals

are the main object in the investigation of ring theory and provide important information

about the rings because they are structural pieces. The same may occur in quantales.

The deﬁnition of prime ideals proposed in [45,46] is based on elements of a quantale and

we ponder it is geared to commutative environment. When we move from commutative

to the noncommutative setting, elementwise should be replaced by an approach based

on ideals. Nevertheless, some authors deﬁned the concept of primeness for commutative

and noncommutative cases without realizing that this concept may not be suitable for

noncommutative setting as it was well shown by Navarro et. al. in [13]. We state that the

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 45

concept of prime ideal of general quantales could be deﬁned as it is done in ring theory,

i.e. based on ideals. The concept of prime ideal provided for quantales by Lingyun Yang

and Luoshan Xu is more suitable for commutative quantales. Therefore, this chapter

provides a new concept of prime ideal for a general (commutative and noncommutative)

quantale which the elementwise prime ideal deﬁnition proposed by Lingyun Yang and

Luoshan Xu is called completely prime ideal.

The ﬁrst aim is to study the notion of primeness in the following perspective: I renamed

prime ideal deﬁned in [45] to completely prime ideal and deﬁne a new concept of prime

ideal for quantales. Then It is translated an important result in ring theory for quantales

environment (theorem 35) to prove that these two concepts coincide in the commutative

setting, but are no longer valid in the noncommutative setting (see Proposition 67).

Besides, based on the studies of Lawrence and Handelman [2], started in 1975 I developed

the notion of strong primeness for general quantales. The second aim is to propose the

concept of fuzzy primeness and fuzzy strong primeness as well as fuzzy uniform strong

primeness for quantales following the ideas developed in previous chapters.

At the end of this chapter, I introduce the initial ideas of t-systems and m-systems for

quantales. As a consequence an ideal is prime iﬀ its complement is an m-system.

6.2 Primeness in Quantales

This section proposes a new concept of prime ideals suitable for commutative and non-

commutative quantales. The deﬁnition of prime ideal used in [45,46] will be called herein

completely prime ideals. We drew attention to the theorem 35 where prime ideals can

be characterized in a certain way via elements. The Proposition 68 shows that in the

commutative case, prime and completely prime concepts coincide, which are no longer

valid in the noncommutative setting according to Proposition 67. Finally, the concept

of quantale prime is proposed.

Deﬁnition 39.[47] A quantale is a complete lattice Qwith an associative binary operation

◦satisfying:

a◦ _

k∈K

bk!=_

k∈K

(a◦bk), _

k∈K

ak!◦b=_

k∈K

(ak◦b)

for all a, b, ak, bk∈Qand k∈K.

A quantale Qis called commutative whenever a◦b=b◦afor a, b ∈Q. In this work we

denote the least and greatest elements of a quantale by ⊥and >respectively. If there

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 46

exists an element ein Qsuch that x◦e=e◦x=xfor all xin Qthe quantale is called

a quantale with identity. In this work we consider quantales with identity.

Deﬁnition 40.[46] Let Qbe a quantale. A non-empty subset I⊆Qis called a right

ideal of Qif it satisﬁes the following conditions:

i) a, b ∈Iimplies a∨b∈I;

ii) for all a, b ∈Q, a ∈Iand b≤aimply b∈I,

iii) for all x∈Qand a∈I, we have a◦x∈I.

Similarly we may deﬁne left ideal replacing (iii) by: (iii’) for all x∈Qand a∈I, we

have x◦a∈I. If Iis both right and left ideal of Q, we call Ia two-sided ideal or simply

an ideal of Q.

Clearly by (ii) ⊥ ∈ I. Also, the set of all ideals of Qis closed under arbitrary intersections.

In Qwe denote the subset I◦J={i◦j∈Q:i∈I and j ∈J}and A∨B={a∨b:a∈

A and b ∈B}. Since the operation ◦is associative, we have (A◦B)◦C=A◦(B◦C).

Also, if Ais an two-sided ideal, then A◦Q, Q ◦A, Q ◦A◦Q⊆A.

As usual, ∨induces an order relation ≤on Qby putting x≤y⇔x∨y=y. Moreover,

≤is a congruence i.e. for every x, y, u, v ∈Qif x≤yand u≤v, then x◦u≤y◦v.

To prove this, we ﬁrst observe that if w≤zthen, for any s∈Q,s◦w≤s◦zand

w◦s≤z◦sbecause z=w∨zimplies s◦z=s◦(w∨z)=(s◦w)∨(s◦z)and

z◦s= (w∨z)◦s= (w◦s)∨(z◦s); now suppose x≤yand u≤v, then x◦u≤y◦u

and y◦u≤y◦v. Hence, x◦u≤y◦vby transitivity.

In what follows we propose a more general deﬁnition of prime ideals which encompasses

commutative and non-commutative quantales.

Deﬁnition 41.A prime ideal in a quantale Qis any proper ideal Psuch that, whenever

I, J are ideals of Qwith I◦J⊆P, either I⊆Por J⊆P.

Deﬁnition 42.A subset Pof a quantale Qis called completely prime ideal if xand yare

two elements of Qsuch that their product x◦y∈I, then x∈Ior y∈I.

As we will see the concept of prime and completely prime ideals are diﬀerent and coincide

whenever Qis commutative.

Proposition 65.If Pis completely prime, then Pis prime.

Proof. Suppose that Pis completely prime and I◦J⊆P, but J6⊆ P, where I, J

are ideals of Q. Thus, there exists j∈Jsuch that j /∈P. For all i∈Iwe have

i◦j∈I◦J⊆P, as Pis completely prime and j /∈P, then i∈P. Therefore I⊆P.

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 47

The Proposition 67 will show that the converse of this Proposition is not true.

Deﬁnition 43.[46] Let Qbe a quantale and A⊆Q. The least ideal containing Ais

called the ideal generated by A, and denoted as hAi.

Clearly, h∅i ={⊥}. If ∅ 6=A⊆Q, then we have the following result.

Proposition 66.[46] Let Abe a non-empty subset of a quantale Q. Then hAi={x∈

Q:x≤Wn

i=1 ai, f or some positive integer n and a1, . . . , an∈A∪(A◦Q)∪(Q◦A)∪

(Q◦A◦Q)}.

We may denote hai=h{a}i and a◦Q={a} ◦ Q.

Lemma 34.hai◦Q⊆ haifor all a∈Q. If there exists an unit 1in Q, then hai ◦Q=hai.

Proof. Let x◦q∈ hai ◦ Q, where x∈ haiand q∈Q. Hence, x◦q≤Wn

i=1(ai◦q), where

ai◦q∈a◦Q∪Q◦a∪Q◦a◦Q. Thus, x◦q∈ hai. On the other hand if there exists

unit 1 in Q, we write z∈ haias z=z◦1. Thus, z∈ hai ◦ Qand we have hai ◦ Q=hai.

Theorem 35.For an ideal Pin Qthe following statements are equivalent:

(1) Pis prime ideal;

(2) hai◦hbi ⊆ Pimplies a∈Por b∈P;

(3) a◦Q◦b⊆Pimplies a∈Por b∈P.

Proof. For (1) ⇒(2) note that haiand hbiare ideals of Q. As Pis prime and hai◦hbi ⊆ P,

then hai ⊆ Por hbi ⊆ P. Hence, a∈Por b∈P. For (2) ⇒(1), assume that I◦J⊆P,

but J6⊆ P, where I, J are ideals of Q. Thus, there exists j∈Jsuch that j /∈P. Given

i∈Iwe have hii ⊆ I. Hence, hii◦hji ⊆ I◦J⊆P. By hypothesis i∈Por j∈P. As

j /∈Pthen we have i∈P. Therefore, I⊆P.

For (3) ⇒(1), assume that I◦J⊆P, but J6⊆ P, where I, J are ideals of Q. Thus,

there exists j∈Jsuch that j /∈P. Given i∈Iwe have i◦Q◦j⊆I◦J⊆P. Hence,

i∈Por j∈P, as j /∈Pthen we have i∈P. Thus, I⊆P.

For (1) ⇒(3), suppose a◦Q◦b⊆P, we ﬁrst shall show that hai ◦ Q◦ hbi ⊆ P.

For this, let x◦q◦y∈ hai ◦ Q◦ hbi, where x∈ hai,y∈ hbiand q∈Q. Hence, by

Proposition 66,x≤ ∨n

i=1aiand y≤ ∨m

j=1bj, where ai∈(a◦Q)∪(Q◦a)∪(Q◦a◦Q)

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 48

and bi∈(b◦Q)∪(Q◦b)∪(Q◦b◦Q). Hence x◦q◦y≤(∨n

i=1ai)◦q◦∨m

j=1bj=

(∨n

i=1(ai◦q)) ◦∨m

j=1bj=∨n

i=1(ai◦q◦ ∨m

j=1bj) = ∨n

i=1 ∨m

j=1(ai◦q◦bj).

Observe that ai∈a◦Q∪Q◦a∪Q◦a◦Qand bj∈b◦Q∪Q◦b∪Q◦b◦Qit is no hard to

see that ai◦q◦bj∈a◦Q◦b⊆Pfor all i, j. As Pis an ideal we have x◦q◦y∈P. Thus,

hai ◦ Q◦ hbi ⊆ P. By the Lemma 34 hai ◦ Q=hai. Then, hai ◦ Q◦ hbi=hai ◦ hbi ⊆ P.

By the ﬁrst proof ((1)⇔(2)) we have a∈Por b∈P.

Proposition 67.There exists a noncommutative quantale where a prime ideal is not a

completely prime ideal.

Proof. Consider Gthe invertible 2×2matrices under multiplication over the real interval

[0,1] and the partial order A≤B⇔aij ≤bij. According to Rosenthal ([47], page 19,

example 16) any complete partially ordered group (written multiplicatively) is a quantale

with a◦b=a·b. Thus, Gis a noncommutative quantale.

Let h0ias an ideal generated by 0, clearly h0i={0}. We will show that the h0i(zero

ideal) is prime, but not completely prime by using the Theorem 35 (3). Thus, suppose

that X= a b

c d !and Y= e f

g h !are two matrices such that X◦G◦Y⊆ h0i.

Hence X◦T◦Y= 0 0

0 0 !for all matrix T∈G. Then, in particular,

X◦ 1 0

0 0 !◦Y= a b

c d ! 1 0

0 0 ! e f

g h != ae af

ce cf != 0 ⇔a=c=

0or e=f= 0,

X 0 1

0 0 !Y= a b

c d ! 0 1

0 0 ! e f

g h != ag ah

cg ch != 0 ⇔a=c=

0or g=h= 0,

X◦ 0 0

1 0 !◦Y= a b

c d ! 0 0

1 0 ! e f

g h != be bf

de df != 0 ⇔b=d=

0or e=f= 0,

X◦ 0 0

0 1 !◦Y= a b

c d ! 0 0

0 1 ! e f

g h != bg bh

dg dh != 0 ⇔b=d=

0or g=h= 0,

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 49

Hence, a solution must verify that X= 0 0

0 0 !or Y= 0 0

0 0 !. Therefore X∈ h0i

or Y∈ h0iand then h0iis prime. Nevertheless, h0iis not completely prime, since

0 1

0 0 !◦ 0 1

0 0 != 0 0

0 0 !although 0 1

0 0 !/∈ h0i.

Proposition 68.In a commutative quantale an ideal is completely prime iﬀ it is prime.

Proof. If Pis a completely prime ideal of a quantale Q, then by the Proposition 65 P

is prime. On the other hand, suppose Pis a prime ideal and a◦b∈Pfor any a, b ∈Q.

Let x◦y∈ hai ◦ hbi, where x∈ haiand y∈ hbi. Thus, x◦y≤Wn

i=1 ai◦Wm

j=1 bj=

i=1 Wm

j=1(ai◦bj), where ai∈a◦Q∪Q◦a∪Q◦a◦Qand bj∈b◦Q∪Q◦b∪Q◦b◦Q.

As Qis commutative a◦Q=Q◦a=Q◦a◦Qand b◦Q=Q◦b=Q◦b◦Q.

Thus, ai◦bj∈a◦Q◦b◦Q=a◦b◦Qfor all i= 1, . . . , n and j= 1, . . . , m. Hence,

ai◦bj=a◦b◦q∈Pand then x◦y≤Wn

i=1 Wm

j=1(ai◦bj)∈P. Therefore, hai◦hbi ⊆ P

and by the Theorem 35 we have a∈Por b∈P.

In what follows, we will introduce the notion of quantale prime. As we know, in ring

theory, a quantale is prime iﬀ the ideal generated by 0is a prime ideal. Then, the next

Proposition translates this result into quantale environment and opens the investigation

on quantales prime.

Deﬁnition 44.A quantale Qis called prime if given a, b ∈Qwith a6=⊥and b6=⊥,

there exists f∈Qsuch that a◦f◦b6=⊥.

Proposition 69.A quantale Qis prime iﬀ h⊥i is a prime ideal.

Proof. Suppose Qprime and assume that I◦J⊆ h⊥i, but I , J 6⊆ h⊥i, where I , J are

ideals of Q. Thus, there exists i∈I,j∈Jsuch that i, j 6=⊥. As Qis prime, there exists

q∈Qsuch that i◦q◦j6=⊥, then we have a contradiction because i◦q◦j∈I◦J⊆ h⊥i.

Hence, I⊆ h⊥i or J⊆ h⊥i. On the other hand, suppose h⊥i is a prime ideal of Q.

Given a, b 6=⊥in Q, suppose a◦q◦b=⊥for all q∈Q. Hence, a◦Q◦b⊆ h⊥i. As h⊥i

is a prime ideal, then a∈ h⊥i or b∈ h⊥i, but a, b 6=⊥.

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 50

6.3 Strong Primeness in Quantales

Strongly prime rings were introduced in 1973, as a prime ring with ﬁnite condition in

the generalization of results on group rings proved by Lawrence in his master’s thesis. In

1975, Lawrence and Handelman [2] came up with properties of those rings and proved

important results, for instance all prime rings may be embedded in a strongly prime

ring; and all strongly prime rings are nonsingular. After such relevant paper, Olson [3]

published a paper about uniform strong primeness and its radical. On the contrary of the

concept of strong primeness, Olson proved that the concept of uniform strong primeness is

two-sided. In this section we bring this concept to quantales making speciﬁc adaptations

for this environment. Finally, it is proposed t- and m-systems for quantales, since it gives

another characterization of prime and uniformly strongly prime ideal.

Deﬁnition 45.Let Abe a subset of a quantale Q. The right annihilator of Ais deﬁned

as Anr(A) = {x∈Q:Ax =h⊥i}. Similarly, we can deﬁne the left annihilator Anl.

Deﬁnition 46.[2] A quantale Qis called right strongly prime if for each x∈Q− {⊥}

there exists a ﬁnite nonempty subset Fxof Qsuch that Anr(x◦Fx) = h⊥i.

Clearly if Qis right strongly prime, then Qis prime. The set Fxis called an insulator

of xin Q.

Proposition 70.If Qis right strongly prime, then every nonzero ideal Iof Qcontains a

ﬁnite subset Fwhich has right annihilator zero.

Proof. Suppose Qright strongly prime. Let x∈Iand x6=⊥and F=x◦Fx⊆I. Thus,

Anr(F) = h⊥i.

It is clear that every right strongly prime quantale is a prime quantale. It is also possible

to deﬁne left strongly prime in a similar manner for right strong primeness.

Deﬁnition 47.A quantale Qis called uniformly strongly prime (usp) if the same right

insulator may be chosen for each nonbottom element.

Proposition 71.A quantale Qis a right uniformly strongly prime iﬀ there exists a ﬁnite

subset F⊆Qsuch that for any two nonbottom elements xand yof Q, there exists

f∈Fsuch that x◦f◦y6=⊥.

Proof. Let Qbe uniformly right strongly prime quantale. Hence Qhas a uniform right

insulator Fwhich is a ﬁnite set such that for any element x∈Q,x◦Fhas no nonbottom

right annihilators. Thus, if xand yare any two nonbottom elements in Q,ycannot be

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 51

in the annihilator of x◦F. Hence there is an f∈Fsuch that x◦f◦y6=⊥. For the

reverse implication it is easy to see that if the condition is satisﬁed then for any x6=⊥

in Q, no nonbottom element annihilates x◦Fon the right. Hence Qis uniformly right

strongly prime

It is clear that the condition in Proposition 71 is not one-sided; consequently, this con-

dition is also equivalent to uniformly left strongly prime, and we have:

Corollary 36.Qis uniformly right strongly prime iﬀ Qis uniformly left strongly prime.

Corollary 37.A quantale Qis uniformly strongly prime iﬀ there exists a ﬁnite subset

F⊆Qsuch that a◦F◦b=⊥implies a=⊥or b=⊥for all a, b ∈Q.

Proof. Straightforward.

Deﬁnition 48.An ideal P6=h⊥i of a quantale Qis called uniformly strongly prime (usp)

ideal if there exists a ﬁnite subset F⊆Qsuch that a◦F◦b⊆Pimplies a∈Por b∈P.

Proposition 72.An ideal Iof a quantale Qis a usp ideal iﬀ there exists a ﬁnite subset

F⊆Qsuch that for any two elements a, b ∈Q\I(complement of Iin Q), there exists

f∈Fsuch that x◦f◦y /∈I.

Proof. Suppose Ia usp ideal of Q. If a /∈Iand b /∈Iby Deﬁnition 48 a◦F◦bis not

a subset of I. Hence, there exists f∈Fsuch that a◦f◦b /∈I. For the converse, note

that by hypothesis it is impossible to have a◦F◦b⊆Iand a /∈Iand b /∈I.

Subsequently we introduce the t-/m-systems. They will give us another characterization

of prime and usp ideals.

Deﬁnition 49.A subset Mof a quantale Qis called an m-system if for any two elements

x, y ∈Mthere exists q∈Qsuch that x◦q◦y∈M.

Deﬁnition 50.A subset Tof a quantale Qis called a t-system if there exists a ﬁnite

subset F⊆Qsuch that for any two elements x, y ∈Tthere exists f∈Fsuch that

x◦f◦y∈T.

Proposition 73.Iis a prime ideal of a quantale Qiﬀ Q\I(the complement of Iin Q)

is an m-system.

Proof. Suppose Ia prime ideal. If a, b ∈R\I, then a, b /∈I. By Proposition 35 the

subset a◦Q◦bis not a subset of I. Thus, there exists q∈Qsuch that a◦q◦b /∈I. For

the converse, let a, b ∈Qsuch that a◦Q◦b⊆I, if aand bnot in I, then a, b ∈Q\I.

By hypothesis there exists q∈Qsuch that a◦q◦b∈Q\I, but a◦Q◦b⊆I.

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 52

Proposition 74.An ideal Iis a usp ideal of a quantale Qiﬀ Q\I(the complement of I

in R) is a t-system.

Proof. This Proposition is similar to Proposition 72.

At the end of section 6.2 we introduced a quantale prime where it was proposed a

right/left strongly quantale prime. Lawrence and Handelman proved that all prime rings

may be embedded in a strongly prime ring. Then, a question arises: based on their

studies,may we have the similar result in quantales?

6.4 Fuzzy Prime and Fuzzy usp Ideals in Quantales

This section introduces the ﬁrst version of fuzzy prime ideals and fuzzy uniformly strongly

prime ideals in quantales compatible with the ideas developed in [13] i.e. a fuzzy concept

on membership function is deﬁned and after that it is proved a coherency with α-cuts.

It is also proved that every fuzzy completely prime ideal is a fuzzy prime ideal but the

converse is not true in noncommutative quantale according to Proposition 67.

The intersection and union of fuzzy sets are given by the point-by-point inﬁmum and

supremum. I will use the symbols ∧and ∨for denoting the inﬁmum and supremum of

a collection of real numbers. Hence, _Ais the supremum of a set Aand ^Ais the

inﬁmum of a set A. Again, x◦Adenotes the set {x◦a:a∈A}and x◦A◦y=

{x◦a◦y:a∈A}.

Deﬁnition 51.[46] Let Qbe a quantale. A fuzzy subset Iof Qis called a fuzzy ideal of

Qif it satisﬁes the following conditions for x, y ∈Q:

(1) if x≤y, then I(x)≤I(y);

(2) I(x∨y)≥I(x)∧I(y);

(3) I(x◦y)≥I(x)∨I(y).

From (1) and (2) it follows that I(x∨y) = I(x)∧I(y). Thus, a fuzzy subset Iis a fuzzy

ideal of Qiﬀ I(x∨y) = I(x)∧I(y)and I(x◦y)≥I(x)∨I(y).

Let µbe a fuzzy subset of Xand let α∈[0,1]. Then the set {x∈X:µ(x)≥α}is

called the α-cut. Clearly, if t > s, then µt⊆µs. Again, it is proved in [46] that Iis a

fuzzy ideal of Qiﬀ Iαis an ideal of Qfor all α∈(I(>),1].

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 53

Deﬁnition 52.A non-constant fuzzy ideal P:Q−→ [0,1] is a fuzzy prime ideal of Qif

for any x, y ∈Q,^P(x◦Q◦y) = P(x)∨P(y).

Deﬁnition 53.A non-constant fuzzy ideal P:Q−→ [0,1] is said to be fuzzy completely

prime (fcp) ideal of Qif for any x, y ∈Q,P(x◦y) = P(x)or P(x◦y) = P(y).

Deﬁnition 54.A non-constant fuzzy ideal I:Q−→ [0,1] is said to be fuzzy uniformly

strongly prime (fusp) ideal if there exists a ﬁnite subset F⊆Qsuch that ^I(x◦F◦

y) = I(x)∨I(y), for any x, y ∈Q. The subset Fis called insulator of Iin Q.

The following Proposition says that the deﬁnition of fuzzy uniformly strongly prime is

coherent with the α-cuts.

Proposition 75.Iis a fuzzy prime ideal of Qiﬀ Iαis a prime ideal of Qfor all α∈

(I(>),1].

Proof. Suppose Ia fuzzy ideal of Q. Let x, y ∈Qsuch that x◦Q◦y⊆Iα. Thus, x◦q◦y∈

Iαfor all q∈Q. As Iis a fuzzy prime, then we have I(x)∨I(y) = ^I(x◦Q◦y)≥α

hence I(x)≥αor I(y)≥αi.e. x∈Iαor y∈Iα. Thus, by Theorem 35 Iαis a

prime ideal. On the other hand, suppose Iαprime ideal of Qfor all α∈(I(>),1] and

^I(x◦Q◦y)> I(x)∨I(y). Let t=^I(x◦Q◦y), and thus t>I(x)∨I(y)and

x, y /∈It, but this is a contradiction because I(x◦q◦y)≥tfor all q∈Qi.e. x◦Q◦y⊆It,

as Itis a prime ideal then x∈Itor y∈It. Therefore, Iis a fuzzy prime ideal.

Proposition 76.[46] For a fuzzy ideal Pin Qthe following statements are equivalent:

(1) Pis fuzzy completely prime ideal;

(2) I(x◦y) = I(x)∨I(y)for all x, y ∈Q;

(3) Iαis completely prime ideal of Qfor all α∈(I(>),1].

Proposition 77.If Pis a completely fuzzy prime ideal of Q, then Pis a fuzzy prime ideal

of Q.

Proof. Use Proposition 75 and Proposition 76.

Proposition 78.In a commutative quantale an ideal is completely fuzzy prime iﬀ is fuzzy

prime.

Proof. Use Propositions 68,75 and 76.

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 54

Theorem 38.For a fuzzy ideal Pin Qthe following statements are equivalent:

(1) Pis a fuzzy prime ideal;

(2) Given x, y ∈Qand Ja fuzzy ideal of Qwe have: J(x◦r◦y)≤P(x◦r◦y)for all

r∈Qimplies J(x)≤P(x)or J(y)≤P(y).

Proof. (1)⇒(2), suppose Pfuzzy prime ideal, if J(x◦q◦y)≤P(x◦q◦y)for all q∈Q,

then ^J(x◦Q◦y)≤^P(x◦Q◦y). As Jis a fuzzy ideal, by Deﬁnition 51 (3) we have

J(x◦r◦y)≥J(x)∨J(r)∨J(y)≥J(x)∨J(y), hence VJ(x◦Q◦y)≥J(x)∨J(y). Thus,

J(x)∨J(y)≤^J(x◦Q◦y)≤^P(x◦Q◦y) = P(x)∨P(y). Hence, J(x)∨J(y)≤

P(x)∨P(y). Therefore, J(x)≤P(x)or J(y)≤P(y). For (2)⇒(1), suppose that

^P(x◦Q◦y)> P (x)∨P(y)for some x, y ∈Q. Then there exists t∈(0,1) such that

^P(x◦Q◦y)> t > P (x)∨P(y). Now, deﬁne the ideal I:Q−→ [0,1] given by:

I(z) = (P(z), if P (z)≥t

t, otherwise

This is a fuzzy ideal with t<I(x◦q◦y) = P(x◦q◦y)for all q∈Q, but t=I(x) =

I(y)> P (x)∨P(y).

Proposition 79.Iis a fusp ideal of Qiﬀ Iαis a usp ideal of Qfor all α∈(I(>),1].

Proof. Suppose Ia fusp ideal and F⊆Qa ﬁnite subset given by Deﬁnition 54. Let

x, y ∈Qand α∈(I(>),1] such that x◦F◦y⊆Iα. Hence, I(x)∨I(y) = VI(x◦F◦y)≥α,

and thus I(x)≥αor I(y)≥α. Therefore, x∈Iαor y∈Iα. On the other hand, suppose

Iαa usp ideal of Qfor all α∈(I(>),1]. According to Deﬁnition 48 each Iαhas a ﬁnite

set Fαsuch that if x◦Fα◦y⊆Iαimplies x∈Iαor y∈Iα. Consider the ﬁnite set

F=Tα∈(I(>),1] Fα. Suppose VI(x◦F◦y)> I(x)∨I(y)and t=VI(x◦F◦y)for

some x, y ∈Q. Note that t > I (x)∨I(y)and t≤I(x◦f◦y)for all f∈F. Hence,

x, y 6∈ It, but x◦F◦y⊆Itand thus (by hypothesis) x∈Itor y∈It, where we have a

contradiction. Therefore, VI(x◦F◦y) = I(x)∨I(y).

Corollary 39.If Pis a fusp ideal of Q, then Pis a fuzzy prime ideal of Q.

Proof. Use Proposition 75 and 79

Chapter 6. Prime ideals and fuzzy prime ideals over noncommutative quantales 55

The next proposition enables us to build a fuzzy ideal from a crisp ideal. Also, it is

another way to verify if a subset Sis an ideal of Qor not.

Proposition 80.Let Jbe an ideal of Qand α∈(I(>),1). Deﬁne I:Q−→ [0,1] as

I(x) = 









1, if x =⊥;

α, if x ∈J\ {⊥};

0, if x /∈J.

Then:

i) Iis a fuzzy ideal of Q;

ii) Iis a fusp ideal iﬀ Jis a usp ideal .

Proof. i) Note that for all α-cut (α∈(I(>),1)) we have Iα=Jand I1={⊥}. Hence,

Iis a fuzzy ideal since Iαand I1are ideals. ii) Suppose Ia usfp ideal and let x, y /∈J.

As Iis usfp, there exists a ﬁnite set F, where VI(x◦F◦y) = I(x)∨I(y)=0. Since

Fis ﬁnite, there exits f∈Fwhere I(x◦f◦y)=0, then x◦f◦y /∈J. On the

other hand, suppose Jis usp ideal of Q. According to Proposition 37 there exists a

ﬁnite subset F⊆Qsuch that a◦F◦b=⊥implies a=⊥or b=⊥for all a, b ∈Q.

Thus, given x, y ∈Qwe have the following cases: 1) If x, y =⊥, then we have triviality

VI(x◦F◦y) = I(⊥) = I(x)∨I(y); 2) If x∈Jor y∈J, then x◦F◦y⊆J. Thus,

VI(x◦F◦y) = α=I(x)∨I(y); 3) If x /∈Jand y /∈J, then by Proposition 72 there

exists f∈Jsuch that x◦f◦y /∈J. Therefore, VI(x◦F◦y) = 0 = I(x)∨I(y).

6.5 Final Remarks

Prime ideals have developed an important role in ring theory and have attracted the

attention of some researchers in the investigation of quantales. As prime ideals are

structural pieces of a ring it is relevant to study its concept in order to establish a well-

founded quantale theory. With this in mind, it is necessary to investigate primeness over

arbitrary quantales, i.e. commutative and noncommutative setting.

Chapter 7

Next Steps

In this work i introduced the m-systems and proved that the fuzzy ideal is prime iﬀ its

complement is an m-system. The next steps must develop this theory for noncommuta-

tive rings and must extend the paper of Navarro [13].

As I have said in the introduction of this work, I can think of uniformly strongly prime

fuzzy ideals in two manners: One, to solve crisp problems and two, to develop a pure

fuzzy ideals theory. With respect to solve crisp problems, a future work may increase

the connections between crisp and fuzzy ideal theory, starting by Proposition 35. It is

necessary to improve this connection if we want to solve crisp problems. Concerning the

development of fuzzy ideal theory we have some conjectures and answers, for instance:

1 - The Conjecture 1in the Chapter 3: If the ideal Kis Maximal, then we have the

maximality principle for uniformly strongly prime fuzzy ideals. On the other hand, if this

conjecture is false, then we have another example which shows the diﬀerence between

fuzzy and crisp algebra theory. The answer of Conjecture 2can extend the Proposition

36 to noncommutative rings.

2 - Zadeh’s Extension is important for us because it can be associated with problems

involving isomorphism theorem. For instance, the Corollary 20 shows how we can solve

problems for a ring Susing the ring R. Thus, the Question 2discusses about the

classes of rings, where the uniformly strongly prime fuzzy ideals are preserved by Zadeh’s

Extension.

3 - As we know the radical theory was very important to understand rings and their

structures. This work introduced the uniformly strongly radical of a fuzzy ideal and

proved some initial results on it. But it is necessary to increase this work by making

connections between the radical and fuzzy ideals. It is also important to decide which

deﬁnition of a fuzzy radical is more appropriate, since this work suggests three concepts.

Chapter 8. Future Works 57

Another issue is the deﬁnition of strongly prime fuzzy ideals given in this work, which

was based on α-cuts. This deﬁnition is not appropriate because it does not produce new

results in a pure fuzzy environment or does not show the diﬀerences between crisp and

fuzzy setting.

The concept of strong primeness introduced in this work brings some questions about

strong primeness for quantale environment. For instance: Is the concept of right and

left strong primeness in quantales distinct? May all prime quantales be embedded in a

strongly prime quantale?

Appendix A

Publications

In this I highlight the published papers with their abstracts. I draw attention to the

fourth work in the next sequence, where the diﬀerence between crisp and fuzzy ring

theory on usp ideals is presented. This paper (see [22]) won the third place as the

best work at NAFIPS 2014, Boston - USA. The papers 8 and 9 do not deal with

Strongly Primeness in the fuzzy environment, they are extra publications.

A.1 Published Studies

1 - Strongly Prime Fuzzy Ideals Over Noncommutative Rings

[20]

Abstract: In this paper it is deﬁned the concept of strongly prime fuzzy ideal for non-

commutative rings. Also, it is proved that the Zadeh’s extension preserves strongly fuzzy

primeness and that every strongly prime fuzzy ideal is a prime fuzzy ideal as well as every

fuzzy maximal is a strongly prime fuzzy ideal.

2 - On Properties of fuzzy ideals [48]

Abstract: The main goal of this paper is to investigate the properties of fuzzy ideals of

a ring R. It provides a proof that there exists an isomorphism of lattices of fuzzy ideals

when ever the rings are isomorphic. Finite-valued fuzzy ideals are also described and a

method is created to count the number of fuzzy ideals in ﬁnite and Artinian rings.

3 - Uniformly Strongly Prime Fuzzy Ideals [23]

Abstract: In this paper we deﬁne the concept of uniformly strongly prime fuzzy ideal for

associative rings with unity. This concept is proposed without dependence of level cuts.

Appendix Publications 59

We show a pure fuzzy demonstration that all uniformly strongly prime fuzzy ideals are

a prime fuzzy ideal according to the newest deﬁnition given by Navarro, Cortadellas and

Lobillo [13] in 2012. Also, some properties about fuzzy strongly prime radical and their

relations with Zadeh’s extension are shown.

4 - A Fuzzy Version of Strongly Prime Ideals [22]

Abstract: This paper is a step forward in the ﬁeld of fuzzy algebra. Its main target is

the investigation of some properties about uniformly strongly prime fuzzy ideals (uspf)

based on a deﬁnition without α-cuts dependence. This approach is relevant because it

is possible to ﬁnd pure fuzzy results and to see clearly how the fuzzy algebra is diﬀerent

from classical algebra. For example: in classical ring theory an ideal is uniformly strongly

prime (usp) if and only if its quotient is a usp ring, but as we shall demonstrate here,

this statement does not happen in the fuzzy algebra. Also, we investigate the Zadeh’s

extension on uspf ideals.

5 - The Strongly Prime Radical of a Fuzzy Ideal [30]

Abstract: In 2013, Bergamaschi and Santiago [20] proposed Strongly Prime Fuzzy(SP)

ideals for commutative and noncommutative rings with unity, and investigated their

properties. This paper goes a step further since it provides the concept of Strongly Prime

Radical of a fuzzy ideal and its properties are investigated. It is shown that Zadeh’s ex-

tension preserves strongly prime radicals. Also, a version of Theorem of Correspondence

for strongly prime fuzzy ideals is proved.

6 - New Types of Strongly Prime Fuzzy Ideal [21]

Abstract: Inspired on the ideas of Malik, Moderson and Navarro about fuzzy primeness,

the current paper goes a step further since it provides a characterization of Strongly

Prime fuzzy ideals. To achieve that, new kind of fuzzy ideals are introduced: Semi-prime,

Primary,Special Strongly Prime (SSP) and Almost Special Strongly Prime (ASSP). The

last two types of ideals have no crisp correspondents in Algebra. All the ideals together

play a fundamental role to prove that crisp results are also valid in the fuzzy environment.

The paper also shows how Zadeh’s extension behaves in such new fuzzy ideals.

7 - On Properties of Uniformly Strongly Prime Fuzzy Ideals

[37]

Abstract: The main purpose of this paper is to continue the study of uniform strong

primeness in fuzzy setting started in 2014. A pure fuzzy notion of this structure allows

us to develop speciﬁc fuzzy results on Uniformly Strongly Prime (USP) ideals over com-

mutative and noncommutative rings. Besides, the diﬀerences between crisp and fuzzy

setting are investigated. For instance, in crisp setting an ideal Iof a ring Ris a USP

Appendix Publications 60

ideal iﬀ the quotient R/I is a USP ring. Nevertheless, when working over fuzzy setting

this is no longer valid. This paper shows new results on USP fuzzy ideals and proves

that the concept of uniform strong primeness is compatible with α-cuts. Also, the t- and

m-systems are introduced in a fuzzy setting and their relations with fuzzy prime and

uniformly strongly prime ideals are investigated.

8 - Fuzzy Quaternion Numbers [49] - Extra Paper

Abstract: In this paper we build the concept of fuzzy quaternion numbers as a natural

extension of fuzzy real numbers. We discuss some important concepts such as their

arithmetic properties, distance, supremum, inﬁmum and limit of sequences.

9 - Rotation of Triangular Fuzzy Numbers via Quaternion

[50] - Extra Paper

Abstract: In this paper we introduced the concept of three-dimensional triangular fuzzy

number and their properties are investigated. It is shown that this set has important

metrical properties, e.g convexity. The paper also provides a rotation method for such

numbers based on quaternion and aggregation operator.

A.2 Unpublished Studies

10 - On Properties of Uniformly Strongly Prime Fuzzy Ide-

als - Full Version

Paper accepted in 10/15/2015: Journal of Communication and Computer, USA.