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Research J. Pharm. and Tech. 12(4): April 2019
1991
ISSN 0974-3618 (Print) www.rjptonline.org
0974-360X (Online)
REVIEW ARTICLE
A Review on Orthogonal Derivations in Rings
Kotte Amaranadha Reddy1, K Madhusudhan Reddy2, S. Sharief Basha3
1Research Scholar, Vellore Institute of Technology, Vellore
2Lecture, Math Section, Information Technology Shinas College of Technology, Sultanate of Oman.
3Assistant Professor, Vellore Institute of Technology, Vellore
*Corresponding Author E-mail: amar.anil159@gmail.com
ABSTRACT:
This paper presents a brief review of derivations used in rings such as orthogonal derivation, orthogonal generalized
derivation, orthogonal Jordan derivation, orthogonal symmetric derivation, and orthogonal semiderivation.
KEYWORDS: Derivations, orthogonal derivation, orthogonal bi-derivation, orthogonal generalized derivation,
orthogonal semi derivation.
AMS Subject Classifications:16W25, 16N60, 16A12, 16N80, 16U70, 16D25, 17B40.
INTRODUCTION
The study of algebraic number theory and ideals had a
great impact on the development of ring theory. Jullus
Wiihelm Richard Dedekind, a famous German
mathematician introduced the concepts of fundamentals
of ring theory though the name ring has been given later
by Hilbert. Dedekind has contributed a lot to abstract
algebra, an axiomatic foundation for the natural
numbers, algebraic number theory and the definition of
the real numbers. In 1879 and 1894 the notions of an
ideal had led to the fundamental of ring theory.
Algebraic structure plays an important role in ring
theory. Some special classes of rings are group ring,
division ring, universal enveloping algebra, and
polynomial identities. These kinds of rings are used in
solving a variety of problems in number theory and
algebra. There are many examples of rings found in
other areas of mathematics which includes topology and
mathematical analysis. Derivation in ring theory was
introduced by E. C. Posner [2] in 1957. In the process of
improving the derivations in ring theory, there are
various derivations such as generalized derivation,
Jordan derivation, symmetric bi-derivation, and
generalized Jordan derivation has been developed.
Received on 11.10.2018 Modified on 18.11.2018
Accepted on 19.12.2018 © RJPT All right reserved
Research J. Pharm. and Tech. 2019; 12(4):1991-1996.
DOI: 10.5958/0974-360X.2019.00333.0
In the year 1989, M. Bresar and J. Vukman introduced
the concepts of orthogonal derivation in a ring. The main
aim of this review article is to present the studies on
orthogonal derivations.
We represent a following chart for several types of
orthogonal derivations in rings.
Orthogonal Derivations
Generalized
Orthogonal
Derivations
Orthogonal
Symmetric bi
Derivations
Orthoonal
Jordn
Derivation
Orthogonal
Semiderivations
PRELIMINARIES:
Definition 1.1
A non-empty set R with two binary operations of
addition and multiplication is said to be a ring if the
following conditions are satisfied
• is an abelian group.
• is a semi-group.
• Multiplication is distributive over addition i.e.
and
for all in
Definition 1.2
A Non-associative ring is an additive abelian group in
which multiplication is defined, which is distributive
over addition, on the left as well on the right, that
is, for all
in .
Research J. Pharm. and Tech. 12(4): April 2019
1992
Definition 1.3 Let be a semiring with two binary
operations, ‘+’ and ‘*’ such that
• is a semigroup.
• is a semigroup.
• The two distributive laws are satisfied. That is, for
all in
and
Definition 1.4 A set be a nearing with two binary
operations ‘+’ and ‘.‘ is near ring, if
• is a group (need not be abelian).
• is a semi group.
• or, for all
in .
Definition 1.5 Let be a non-empty subset of ring
with the property that is a subgroup of additive group
then
• is a right ideal in if is closed under
multiplication on the right by the elements of i.e.
for each and
• is a left ideal in if is closed under
multiplication on the left by the elements of i.e.
for each and
• is an ideal in if it is both a right ideal as well as
a left ideal in i.e. for each and
and,
Definition 1.6 A ring R is called prime if
implies or , for all in
Definition 1.7 A ring is called semi prime if
implies, for all in .
Definition 1.8 The Center of is defined as
.
Definition 1.9 An additive mapping d from a ring R to R
is called a right derivationif ,
for all in
Definition 1.10 An additive mapping from to is
called left derivation if for
allin .
Definition 1.11 An additive mapping from to is
called derivation if it is both right and left derivations.
Example: Consider
, where
is a semiring.
Define a mapping by
,
for all .
Definition 1.12 An additive map is called a
reverse derivation if for all
Example: Let
, where
is a semiring. Define a mapping by
is a reverse derivation.
Definition 1.13 Let be an associative ring. An additive
mapping is called a semi derivation
associated with a function if, for all in
•
• f(g(x)) = g(f(x)).
Example: Let be a semiring. Let
, define a
mapping by
, for all
.
by
, for all
.
Definition 1.14 An additive mapping is called
semiderivation associated with a function g
satisfying for all
in is called a reverse semi derivation.
Example: Let
, where is a
semiring.
Define a mapping by
,
for all .
by
, for all .
Definition 1.15 An additive mapping is said to
be a right generalized derivation if there exists a
derivation from to such that
for all in .
Definition 1.16 An additive mapping is said to
be a left generalized derivation if there exists a
derivation from to such that
, for all in.
Definition 1.17 An additive mapping is said to
be a generalized derivation if it is both right and left
generalized derivations.
Example: Let
. If we define
the mapping ,
Such that by
and by
, Then is a generalized
derivation of with associated derivation but not a
Research J. Pharm. and Tech. 12(4): April 2019
1993
derivation of
Definition 1.18 An additive mapping is called
right Jordan derivation if , for
all in
Definition 1.19 An additive mapping is called
left Jordan derivation if , for all in
.
Definition 1.20 If is a Jordan derivation if it
is both right and left Jordan derivations.
Example: Let be a ring and such that
for all and for some in
Define a map by Then it is
very easy to see that is a Jordan derivation on but
not a derivation on
Definition 1.21 A bi-additive mapping is
called a bi-derivation if
and , for all
in .
Definition 1.22 A symmetric bi-additive mapping
is called a symmetric bi-derivation
if for all
in.Obviously, in this case also the relation
for allin.
Example: The map defined by
is symmetric map.
, if
Then is a Symmetric Bi - derivation
Definition 1.23 Two derivations are called
orthogonal if for all
Example: Let Define the operation +
and * on as follows
Then is semiring Define such that
such that
In this review article we give brief introduction about
derivation, generalized derivation and Jordan
derivations.
DERIVATIONS:
[1] Nobuo Nobusawa in the year 1964 wrote the
definitions and its examples for ring, simple
ring, semisimple ring, operator rings ideal and -
sub modules. In [2] EdwardC. Posner introduced the
concepts of derivations in 1957and proved two theorems.
The first theorem says if is a prime ring of
characteristics not 2, if the iterate of two derivations is a
derivations, then one of them is zero. The second
theorem says that if - be the prime ring and is a
derivation of a prime ring such that, for all
is in the center of Then if is the
zero derivation, is commutative. [3] Nurcon Argac et.
al., in 1987 defined which represents a prime ring with
center and represents a ring automorphism of
Further he denotes be a derivation of
namely on additive endomorphism of such that
for all [4] In the
year 2010 the generalized derivations on semirings was
introduced by M.Chandramouleswaran and V.
Thiruveni. They also investigated some interesting
results on commutativity by using generalized
derivations.
BI – DERIVATIONS:
In 1989, J Vukuman [5] verified the results of E.C
Posner which states if is a prime ring of characteristic
not two and are non zero derivations on then the
mapping cannot be a derivation. After a
year the same author [6] proved some two results
concerning symmetric bi – derivation on prime and
semiprime ring can be found in J Vukuman [5]. The first
results says that, if and symmetric bi – derivations
on prime ring of characteristic different from two and
three such that holds for all
then either or The second results
proved that, the existence of a non zero symmetric bi-
derivation on prime ring of characteristic different from
two and three, such that for all
In 1993, M.Serif Yenigul and NurcanArgac[7]verified
the results of J Vukuman [5]by concering symmetric bi –
derivation on prime and semiprime rings. They verified
the results by replacing associative ring for a non zero
ideal of
Research J. Pharm. and Tech. 12(4): April 2019
1994
GENERALIZED DERIVATIONS:
In [8] T. Siukwen Lee in the year 1999, studied the
extension problem of generalized derivations and hence
presented the characterization of generalized derivations.
Hence he provided the exact result of generalized
derivations with nilpotent values on Lie ideals or one
side ideals.[9] In 2007 Bojan Hvala defined generalized
Lie derivation on rings and proved that every generalized
Lie derivation on a prime ring is a sum of generalized
derivation from into its central closure . In 2007[10]
Mehmet Ali Ozturk and Hasret Yzarli introduced
Modules over the generalized centroid of semiprime
ring. In [11] Mohanmad Ashraf et.al in the year
2007, motivated by Ashraf and Rehman[49] and proved
that a prime ring with a non zero ideal must be
commutative if it admits a derivation satisfying either
of the properties or
for all in The authors extended these
results into the commutativity of a primering in which
the generalized derivation satisfies any one of the
following properties, (1) ,
,
,
for all in In [12]
Wu Wei and Wan Zhaoxun in the year 2011, they
presented the concepts of generalized derivations covers
derivations and generalized inner derivations. In [13] C
Jaya Subba Reddy et.al, in 2015, some ideas from
reverse derivation towards the generalized reverse
derivations on semiprime rings. In[14] Nadeem Ur
Rehman et. al., in 2016 provided the generalized results
on commutativity of rings with derivations and on prime
and semiprime rings. They further examined the results
if a ring satisfied the identity.
JORDAN DERIVATIONS:
In [15] Atsushi Nakajma introduced a notion of
generalized Jordan derivation in 2001 and showed that
there is a relation between derivations and corresponding
homomorphism. He provided the set of all generalized
jordan derivations with some categorical properties from
to bi module Finally he proved some results
of Jordan derivations that are easily extended to
generalized jordan derivations on 2 – torsion free
semiprime ring. In [16] Yilmaz Ceven and M Ali Ozturk
in 2004 defined the generalized derivation and a jordan
generalized derivation on rings and showed that a
jordan generalized derivation on some rings is a
generalized derivation. In [17] Mohammad Nagy et.al, in
2010 had proved the reverse jordan and left derivation in
rings. He provided following two theorems. The first
result was that anon zero reverse bi derivation makes a
prime ring commutative and this reverse bi - derivation
becomes an ordinary bi – derivation and the second
results was that a prime ring that admits
a non zero jordan left bi - derivation is commutative. In
[18] Mohamad Ashraf et.al, in 2006 had provided a
historical survey on derivations derivations,
generalized Jordan derivation in rings. They further
provided some applications of derivations. In [19] Alev
Firat, in 2006 introduced the notion of a semiderivation
and he proved generalized some properties of prime
rings with derivations to the primerings with semi-
derivatives.
Now we give brief review about the orthogonal
derivations.
ORTHOGONAL DERIVATIONS:
In [20] Nishteman N. Suliman and Abdul Rahman H.
Majeed in 2012 generalized some results concerning
orthogonal derivations for a non zero ideal of
semiprime ring. Which is the extension of the results
of M. Ashraf et. al, [49]. These results are related to
some results concerning product of derivations on
rings.
Results: See [20]
In [21] in 2014, N. Suguna Thameen and M.
Chandramouleaswran introduced orthogonal derivations
on semirings and they proved some results on semiprime
semirings.
Results: See [54]
In [22] in 2015, N. SugunaThameen and M.
Chandramouleaswran introduced orthogonal derivations
and orthogonal generalized derivations on ideals of
semirings.
Results: See [22]
In [23] U. Revathy et.al, in 2015 introduced the notion of
orthogonality of two reverse derivations on
semiprimesemirings and proved several necessary and
sufficient condition for two derivations to be orthogonal.
Results: See [23]
In [24] Ali Al Hachami KH in 2017, proved few
outcomes concerning two remaining deductions on a
semi prime ring are displayed.
Results: See [24]
In [25] Shakir Ali and Mohammad Salahuddian Khan in
2013, some known results of orthogonal derivations and
orthogonal generalized derivations of semiprime ring
are extended to orthogonal derivations and
orthogonal generalized derivations.
Results: See [25]
ORTHOGONAL BI-DERIVATION:
In [26] M. A. Ozturk and M. Sapanc in 1997 derived the
concept of orthogonal symmetric bi-drivation on
semiprime gammarings.
Results: See [26]
Research J. Pharm. and Tech. 12(4): April 2019
1995
In [27] M. N. Daif et.al, in 2010 presented the notation
of orthogonality between the derivation and bi -
derivation of a ring. They provided the four conditions
equivalent to the notations of orthogonality in the
context 2- toursion free semiprime ring. Further they
provided the orthogonality in terms of non zero ideal of a
2-torsion free semiprime ring.
Results: See [27]
In [28] C. Jaya Subba Reddy and R.Ramoorthy Reddy
introduced the notation of orthogonal symmetric bi –
derivations in semiprime ring in 2016.
Results: See [28]
In the same year same authors proved some results on
orthogonality of derivations and bi
derivations in semiprime rings. These results extended to
orthogonal symmetric bi – derivations in semiprime
rings [29].
Results: See [29]
In 2017 the same authors [30] extended the results of
orthogonal symmetric bi – derivations in semiprimering
toorthogonality conditions for two generalized
symmetric bi – derivations of semiprime rings [28].
Results: See [30]
ORTHOGONAL GENERALIZED DERIVATIONS:
In [31] NurcanArgac et.al in 2004, Prove some results
concerning two generalized derivation on a semiprime
rings and also extended the results of M Bresar and
Vukman [45] to orthogonal generalized derivations.
Results: See [31]
In [32] Emine Albas in 2007, extended the results of
Bresar and J.Vukman [45] to orthogonal generalized
derivations on a non zeron ideal of .
Results: See [32]
In [34] MehsinJabelAlteya in 2010 had investigated
some results on concerning a non zerogeneralized
derivation with left cancellation property on semiprime
ring.
Results: See [34]
In [35] Nishteman N. Suliman et.al, in 2012 studied the
concepts of some results concerning orthogonal
generalized derivations on a semiprime rings.
Results: See [35]
In [36] Salah M. Salih and Hussien J. Thhub in 2013
extended the results of E.C. Posner [2] to the orthogonal
generalized higher derivations on M and obtained
parallel results.
Results: See [36]
In [37] Salah Mehdi Salih in 2013 extended the results of
Bresaret. al [35] to the concepts of orthogonal
derivations and orthogonal generalized derivations on
Module.
Results: See [37]
In [38] N. SugunaThameen and M.
Chandramouleaswran in 2015 extended the results of
orthogonal generalized derivations on semirings and they
proved some results on semiprimesemirings.
Results: See [54]
[39] Cheng Chen Sun is extended to the results of
NurcanArga et.al [31] of the composition of a couple of
generalized (, ) – derivations on a non-zero ideal of a
semiprime ring.
Results: See [39]
ORTHOGONAL SEMI DERIVATIONS:
In [40] K.KanakSindhuet. al, in 2015 defined orthogonal
semiderivations on semiprimesemirings. They
investigated some necessary and sufficient conditions for
two semiderivations to be orthogonal.
Results: See [40]
In [41] U. Revathy et.al, in 2015, Introduced the notion
of orthogonality of two reverse semi derivations on semi
prime semi ring and presented several necessary and
sufficient conditions for two reverse semi derivations to
be orthogonal.
Results: See [41]
In [42] Kyung Ho Kim and Yon Hoon Lee in 2017
introduced the notion of orthogonal reverse
semiderivation on semirings and also investigated the
conditions for two reverse semiderivations on semiring
to be orthogonal.
Results: See [54]
In [43] K. KanakSindhuet. al, in 2015 extended some
orthogonal generalized derivation of semiprimesemirings
to orthogonal generalized semiderivations of
semiprimesemirings.
Results: See [43]
In [44] U.Revathy et.al, in 2016 extended the properties
of orthogonal generalized semiderivation of
semiprimesemiring with left cancellation property on
semiprimesemiring.
Results: See [44]
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