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.
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.
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 .
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 all
in .
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
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 all
in.
Example:
The
map defined by 
is symmetric map.
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, 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
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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
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]
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 (𝜃, ∅) – 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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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