On properties of Uniformly Strongly Prime fuzzy ideals

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 commutative and noncommutative rings. Besides, the differences between crisp and fuzzy setting are investigated. For instance, in crisp setting an ideal I of a ring R is a USP 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 Nevertheless, when working over fuzzy setting this is no longer valid.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.

Full text

On Properties of Uniformly Strongly Prime Fuzzy
Ideals
Flaulles B. Bergamaschi, Regivan H. N. Santiago
Departamento de Inform´
atica e Matem´
atica Aplicada
Universidade Federal do Rio Grande do Norte
Natal, Rio Grande do Norte - Brasil
Email: flaulles@yahoo.com.br, regivan@dimap.ufrn.br
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
commutative 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 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fuzzy setting this is no longer valid.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.
I. INTRODUCTION
Our studies begin with the following question: Is it possible
to have a fuzzy version of a uniform strong prime (USP) ideal?
In 2014, we answered this question by introducing in [1]
the notion of USP fuzzy ideals without α-cut dependence for
the first time. Since then we have been dedicated to investigate
its properties and similarities with crisp setting. So this paper
increases our understanding on USP ideals and fills the gap
related with α-cuts as well as it introduces the concept of
systems for a fuzzy environment.
The following paragraphs introduce the whole subject.
The importance of prime numbers for pure mathematics
results and many applications is well-known. For example,
nowadays prime numbers are the fundamental idea behind
cryptography. We may think of primeness for sets and subsets.
In other words, it is perfectly possible to define prime rings
and prime ideals1.
A ring is a set endowed with two operations, addition and
multiplication, where these operations have properties to those
operations defined for the integers. Prime ideals are subsets
of a ring with similar properties of prime numbers and they
develop an important role in the study of rings as well as prime
numbers in the ring of integers.
Prime rings are a special class of rings where the zero ideal
(0) is a prime ideal. In a commutative ring theory prime rings
are well-known as an integral domain i.e. rings where ab = 0
implies a= 0 or b= 0.
1If the reader has interest in applications about prime ideals and fuzzy
prime ideals in coding theory it is worth reading the introduction of [2] and
its references.
The concept of Strongly Prime (SP) ring appeared in 1973
made by Lawrence in his Master’s thesis as a subclass of
prime rings. In 1975, Lawrence and Handelman [3] presented
an extensive work on SP rings. They developed properties and
proved important results, e.g all prime rings may be embedded
in an SP ring; all SP rings are nonsingular. Besides, they
defined the Uniformly Strongly Prime (USP) ring as a subclass
of SP rings and demonstrated some initial uniform properties.
After a little time, the study of USP rings became more
important. Thus, in 1987, Olson [4] presented a paper about
USP rings and USP radical. Olson proved that USP rings
generate a radical class which properly contains both the right
and left SP radicals and which is independent of the famous
Jacobson and Brown-McCoy radicals. Also, some results in
group ring theory was rediscovered by using the SP and USP
ideals.
In 2013, motivated by crisp problems in group ring theory
(e.g. isomorphism problem) we [5] proposed the concept of
SP ideal for fuzzy environment for the first time. The main
goal was to investigate this structure in fuzzy environment.
Thus, the concept of SP fuzzy ideal was born and it was called
SPf ideal. The difficulty to find a pure fuzzy definition of an
SP ideal forced us to define SPf ideals on α-cuts. But this
approach had some issues. One of them lies in the fact that all
results have similarities in crisp algebra. In other words, fuzzy
setting became a mirror of classical theory.
Although we have not found a pure definition of SPf
ideals yet, the ideas developed in [5] enabled us to create the
definition of USPf (uniformly strongly prime fuzzy) ideal. The
concept of USPf was possible because the crisp definition of
USP ideal is more suitable to be translate to fuzzy environment.
This approach became more interesting, since it allowed us to
find pure fuzzy results. For instance, it was proved that every
USPf ideal is a prime ideal according to the newest pure fuzzy
definition of fuzzy prime ideal given by Navarro [2] in 2012.
Also, it was proved that some results in fuzzy setting are not
true like in classical theory. Besides, the behavior of Zadeh’s
extension on USPf ideals was studied and as a consequence
we built a version of correspondence theorem for USPf ideals.
This paper expands the last three papers [5], [1] and [6]
with new properties of USPf ideals and introduces the t-
and m-systems for a fuzzy setting. It is shown that every
fuzzy ideal is contained in a USPf ideal. Also, it is proved
that the complement of prime/USP fuzzy ideal is an m/t-
system. Moreover the concept of m-systems is compatible

with the newest (2012) definition of fuzzy prime ideals over
noncommutative rings given by Navarro et. al. in [2]. It is
also shown that the inverse image of a USPf ideal is a USPf
ideal when the mapping is a ring homomorphism extended
by Zadeh’s principle. However, on the direct image this is
no longer valid. Finally, we leave for the reader some open
questions about USP investigation on fuzzy setting.
This paper has the following structure: section 2, which
not only provides an overview about the ring and fuzzy
ring theory, but it also contains the classical definition and
results of USP rings/ideals; section 3 has the definition of
USPf ideals and the compatibility with α-cut; section 4 shows
that Zadeh’s extension under epimorphisms does not preserve
uniform strong primeness; section 5 deals with t- /m-systems
and its relations with primeness and uniform strong primeness;
section 6 provides the final remarks.
II. PRELIMINARIES
This section explains some definitions and results that will
be required in the next sections. All rings are associative with
identity and are usually denoted by R. Here, we expose prim-
itive definitions of 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·cand 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 R
is called a right ideal of Rif:
(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: IJ ={i1j1+· · · +
injn:ik∈I and jk∈J, k = 1, . . . , n;where n ∈Z+}
and the set 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 IJ ⊆P, either I⊆Por J⊆P.
Proposition 1 ([7], 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.
Definition 5: 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 6: [3] 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. Handelman and
Lawrence worked exclusively with rings with multiplicative
identity. However, Parmenter, Stewart and Wiegandt [8] have
shown that it is equivalent to:
Definition 7: 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 ([3], Example 1).
Example 1: 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 < n
and 0< q < n. Hence, [pq] = 0 in 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.
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 uni-
formly 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 was 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 [3], Example 1). However, Olson [4] showed that
the concept of uniformly strongly prime ring is two-sided in
view of the following result:
Lemma 1: [4] 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 2: [4] Ris uniformly right strongly prime ring
if and only if Ris uniformly left strongly prime ring.
Lemma 3: [4] 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;
Definition 10: An ideal Iof a ring Ris called USP ideal
if there exists a finite set F⊆Rsuch that for two nonzero
elements xand yof R\I(the complement of Iin R), there
exists f∈Fsuch that xfy /∈I.
Proposition 2: An ideal I of a ring Ris a USP ideal iff
the quotiente R/I is a USP ring.
Proposition 3: An ideal Iof a ring Ris a USP iff there
exists a finite set F⊆Rsuch that xF y ⊆Iimplies x∈Ior
y∈I, where x, y ∈R.
For the following definition consider ∧a t-norm and ∨a
t-conorm.
Definition 11: A fuzzy subset I:R−→ [0,1] of a ring R
is called a fuzzy subring of Rif for all x, y ∈Rthe following
requirements are met:
1) I(x−y)≥I(x)∧I(y);
2) I(xy)≥I(x)∧I(y);
If condition 2) is replaced by I(xy)≥I(x)∨I(y), then I
is called a fuzzy ideal of R.
Note that using properties of the t-norm ∧we have for any
fuzzy subring/fuzzy ideal Iof a ring Rthe following: if for
some x, y ∈R,I(x)< I(y), then I(x−y) = I(x) = I(y−x);
Also, if Iis a fuzzy ideal of a ring R, then I(1) ≤I(x)≤I(0)
for all x∈R.
Proposition 4: [9] A fuzzy subset Iof a ring Ris a fuzzy
ideal of Riff all α-cuts Iαare ideals of R.
Definition 12 (Zadeh’s Extension): Let fbe a function
from set Xinto Y, and let µbe a fuzzy subset of X. Define the
fuzzy subset f(µ)of Yin the following way: 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
Xby f−1(λ)where f−1(λ)(x) = λ(f(x)).
Definition 13: [2] 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 5: [2] 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).
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).
Given an ideal (crisp or fuzzy) of Rdefine R/I ={r+
I:r∈R}the quotient ring by I. In R/I we can define +,·,
where (r+I)+(s+I) = (r+s) + Iand (r+I)·(s+I) =
(rs) + I.
Proposition 6: [4] If Iand Pare ideals of a ring Rwith
Pa USP ideal, then I∩Pis a USP ideal.
III. UNIFORM STRONG PRIMENESS
This section introduces the concept of Uniformly Strongly
Prime fuzzy ideal (or shortly USPf ideal) according to defini-
tion given in [1].
Definition 14: [6] Let Rbe an associative ring with unity.
A non-constant fuzzy ideal I:R−→ [0,1] is said to be
Uniformly Strongly Prime fuzzy (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 7: Let Rbe an associative ring with unity and
Aa crisp USP ideal of R. Then:
I(x) = 1if x ∈A,
0otherwise
is a USPf ideal of R.
Proof: Clearly Iis a fuzzy ideal. According to definition
10 given x, y ∈Rthere exists a finite set F⊆Rsuch that if
xF y ⊆Aimplies x∈Aor y∈A. Suppose xF y ⊆A, then
VI(xF y) = 1 = I(x)∨I(y). If xF y 6⊆ A, then there exists
f∈Fsuch that xfy /∈A. Hence, x, f, y /∈A. Therefore,
VI(xF y) = 0 = I(x)∨I(y).
Proposition 8: 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 14. 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 R
for all I(1) < α ≤I(0). According to proposition 3 each
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 ⊆Itand thus (by hypothesis)
x∈Itor y∈It, where we have a contradiction. Therefore,
VI(xF y) = I(x)∨I(y).
Corollary 4: If Iis a USPf ideal of a ring Riff R/Iαis
a USP ring for all I(1) < α ≤I(0).
Corollary 5: If Iis USPf ideal, then Iis fuzzy prime ideal.
For the following result Ker(f) = {x∈R:f(x) = 0}
is the kernel of the homomorphism fand f−1(J)is the fuzzy
subset of Rby Zadeh’s extension.
Proposition 9: [1] If f:R−→ Sis a homomorphism of
rings and JUSPf ideal of S, then f−1(J)is USPf ideal of R
which is constant on Ker(f).
After proposition 9 we can think about the direct image.
In other words, if Iis a USPf ideal of Rwhich is constat
on Ker(f), then f(I)is USPf is an ideal? In this paper we
proved this statement as false, according to proposition 13 in
the next section.
For the next result consider I∗=II(0) ={x∈R:I(x) =
I(0)}.
Proposition 10: [1] If Iis a USPf ideal of a ring R, then
R/I∗∼
=R/I.
Corollary 6: [1] If f:R−→ Sis an epimorphism and I
USPf ideal of Rwhich is constant on Ker(f), then R/I ∼
=
S/f(I).
Proposition 11: [1] Let Jbe a crisp ideal 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 a USPf ideal iff Jis USP ideal .
Corollary 7: Let Ibe a non-constant fuzzy ideal of R
and define M(x) = (I(0), if x = 0;
α, if x ∈I∗\ {0};
I(1), if x /∈I∗.
Then, Mis a USPf ideal of Riff I∗is a USP ideal of
R.
Corollary 8: Let Ia fuzzy ideal of a ring Rand I m(I)a
finite set. Then, Iis a USPf ideal iff Iαis a USP ideal (crisp)
for all I(1) < α ≤I(0).
Example 2: 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 11. Note that
I∗= (0) is USP ideal. Hence, R/I∗is a USP ring. Applying
the proposition 10 R/I ∼
=R/I∗. Therefore, R/I is a USP
ring, although Iis not a USPf ideal.
The next example show us that direct image of USPf ideals
by Zadeh’s extension on homomorphism are not USPf ideals.
Example 3: [1] 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 a USPf ideal
of Z4, since I1/2=Z4is not a USP ideal.
IV. UNIFORM STRONG PRIMENESS UNDER
HO MO MO RP HI SM
This section amplifies results about USPf ideals. The first
one (proposition 12) is geared to commutative rings. But it
may be valid in noncommutative case (conjecture 2). The
proposition 3 shows the difference between crisp and fuzzy
setting by showing the behavior of Zadeh’s extension on USPf
ideals.
Proposition 12: 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 M
of Rcontaining I∗. Now we can define the following fuzzy set:
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 definition of fuzzy maximal
ideal given by Malik in [11] , Kin the demonstration of
proposition 12 is a fuzzy maximal ideal.
Conjecture 2: The propostion 12 can be extended to non-
commutative rings.
Proposition 13: Let f:R−→ Sis a epimorphism of
commutative and non-USP rings. If Iis a USPf ideal of R
which 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
10 and Corollary 6 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 a USPf ideal.
Question 1: The proposition 13 show us that USPf ideals
can not preserved by Zadeh’s extension. Thus, we ask: Under
which conditions can Zadeh’s extension preserves the USPf
ideals?
Proposition 14: If Iand Pare fuzzy ideals of a ring R
with Pis a USPf ideal, then I∩Pis a 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 15: 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).
The following 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 16: Let Iis a non-constant 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 Ik
is not a USP ideal. Hence, Iis not a USPf ideal.
Proof: When Ik=Ris trivial. Now suppose Ik6=R
and note that in a integral domain if Iis a USP ideal, then I
is a Maximal ideal. Thus, It⊂Ikis impossible, since Itis
Maximal. If Ik⊂Itimplies Iknot maximal, then Iknot a
USP ideal.
V. TH E FU ZZ Y M-A ND T-SYSTEMS
An m-system is a generalization of multiplicative systems.
In the ring theory a set Mis an m-system if for any two
elements x, y in Mthere exists rin Rsuch that the product
xry belongs to M. It is not hard to perceive that an ideal
is prime iff its complement is an m-system (see Mccoy [10]).
On the other hand we have the t-systems which are sets where
given any two elements x, y in Tthere exists a finite set F
such that xfy belongs to Tfor some fin F. Clearly a t-
system is an m-system. Olson [4] proved that Iis a uniformly
strongly prime ideal iff its complement is a t-system.
In this section we will introduce the m-systems in a fuzzy
setting based on the definition of fuzzy prime ideals defined by
Navarro [2] in 2012. The fuzzy t-systems are also introduced.
As we shall soon see it is possible to prove that an ideal I
is a fuzzy prime ideal iff its complement is an m-system.
Moreover, Iis USPf ideal iff its complement is a t-system.
Definition 15: [10] A subset Mof a ring Ris called an
m-system if for any two elements x, y ∈Mthere exists r∈R
such that xry ∈M.
Definition 16: [4] A subset Tof a ring Ris called a t-
system if there exists a finite set F⊆Rsuch that for any two
elements x, y ∈Tthere exists f∈Fsuch that xfy ∈T.
It is not hard to prove that Fis unique. So, it will be called
the insulator of T. Note that, the empty set will be a t-system.
Proposition 17: [10] If Mis a t-system, then Mis a m-
system.
Proposition 18: [10] Iis a prime ideal of a ring Riff R\I
(the complement of Iin R) is an m-system.
Proposition 19: [4] An ideal Iis a USP ideal of a ring R
iff R\I(the complement of Iin R) is a t-system.
For the following definition consider xRy ={xry :r∈
R}.
Definition 17: Let Rbe an associative ring with unity. A
non-constant fuzzy set K:R−→ [0,1] is said to be a fuzzy
m-system if WK(xRy) = K(x)∧K(y), for any x, y ∈R.
Proposition 20: If Kis a fuzzy subset of a ring Rsuch
that Kαis an 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 an
m-system and x, y ∈Ktthen there exists r∈Rsuch that
xry ∈Kti.e K(xry)≥t. Hence, WK(xRy)≥t.
Question 2: Under which conditions may we have the
following result: Kis a fuzzy m-system of Riff Kαis a
m-system for all α-cuts?
For the following results consider Pthe fuzzy ideal and
Pc= 1 −Pthe complement of Pin R.
Corollary 9: Pis a fuzzy prime ideal of Riff Pcis a
fuzzy m-system.
Proof: Suppose Pfuzzy prime, 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). Suppose now Pc
is a 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 following definition consider the subset xF y =
{xfy :f∈F}of ring R.
Definition 18: Let Rbe an associative ring with unity. A
non-constant fuzzy set M:R−→ [0,1] is said to be a fuzzy t-
system if there exists a finite subset Fsuch that WM(xF y) =
M(x)∧M(y), for any x, y ∈R.
Proposition 21: Iis a USPf ideal of Riff Ic(the comple-
ment of Iin R) is a fuzzy t-system.
Proof: Suppose IUSPf, then there exists a finite set F
where 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). Suppose
now Icis a fuzzy t-system, then there exists a finite set F
where 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 22: If Mis a fuzzy t-system of R, then Mα
is 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, Mαis a t-system.
Question 3: Under which conditions may we have the
following result: if Tis a fuzzy t-system of R, then Tis
an m-system?
Example 4: Let Rbe the ring 2×2matrices over the real
numbers. Consider the fuzzy ideal
P(x) = 1if x is the zero matrix,
0otherwise.
By [5] example 1, the zero ideal is prime. Therefore Pis
a fuzzy prime ideal. By corollary 9 the complement Pcof P
in Ris a m-system.
VI. CONCLUSION
Prime ideals are structural pieces of a ring and should be
the first part in the study of its properties. As it is known we
can decompose an ideal in the product of prime ideals. Thus,
the following question is immediate: may we decompose a
fuzzy ideal in a product of fuzzy prime ideals? To answer this
question we need to first of all understand primeness in fuzzy
setting. So, this paper contributes to this end and also develops
some thougths on fuzzy ring theory.
ACKNOWLEDGMENT
The authors would like to thank UESB (Southwest Bahia
State University) and UFRN (Federal University of Rio Grande
do Norte) for their financial support.
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