Jordan Γ*-Derivation on Semiprime Γ-Ring M with Involution
Vol.06No.02(2016), Article ID:67046,11 pages
10.4236/alamt.2016.62006
Ali Kareem1, Hajar Sulaiman1, Abdul-Rahman Hameed Majeed2
1School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia
2Department of Mathematics, University of Baghdad, Baghdad, Iraq
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 21 March 2016; accepted 30 May 2016; published 2 June 2016
ABSTRACT
Let M be a 2-torsion free semiprime G-ring with involution satisfying the condition that 
(
and
). In this paper, we will prove that if a non-zero Jordan G*-derivation d on M satisfies 
for all 
and
, then
.
Keywords:
G-Ring M with Involution, Jordan G*-Derivation, Commutative G-Ring
1. Introduction
The notion of G-ring was introduced as a generalized extension of the concept on classical ring. From its first appearance, the extensions and the generalizations of various important results in the theory of classical rings to the theory of G-rings have attracted a wider attention as an emerging field of research to enrich the world of algebra. A good number of prominent mathematicians have worked on this interesting area of research to develop many basic characterizations of G-rings. Nobusawa [1] first introduced the notion of a G-ring and showed that G-rings are more general than rings. Barnes [2] slightly weakened the conditions in the definition of G-ring in the sense of Nobusawa. Barnes [2] , Luh [3] , Kyuno [4] , Hoque and Pual [5] - [7] , Ceven [8] , Dey et al. [9] [10] , Vukman [11] and others obtained a large number of important basic properties of G-rings in various ways and developed more remarkable results of G-rings. We start with the following necessary introductory definitions.
Let M and G be additive abelian groups. If there exists an additive mapping 
of 
which satisfies the conditions:
1)
,
2)
,
3)
, then M is called a G-ring [2] . Every ring M is a G-ring with M = G. However a G- ring need not be a ring. Let M be a G-ring. Then M is said to be prime if 
with
, implies 
or 
and semiprime if 
with 
implies
. Furthermore, M is said to be a commutative G-ring if 
for all 
and
. Moreover, the set 
is called the center of the G-ring M. If M is a G-ring, then 
is known as the commutator of x and y with respect to
, where 
and
. We make the following basic commutator identities:
(1)
(2)
for all 
and
. Now, we consider the following assumption:
A.......
, for all 
and
.
According to assumption (A), the above commutator identities reduce to 
and
, which we will extensively used.
During the past few decades, many authors have studied derivations in the context of prime and semiprime rings and G-rings with involution [11] - [14] . The notion of derivation and Jordan derivation on a G-ring were defined by [15] . Let M be G-ring. An additive mapping 
is called a derivation if 
for all 
and
. An additive mapping 
is called a Jordan derivation if 
for all 
and
.
Definition 1 [16] . An additive mapping 
on a G-ring M is called an involution if 
and 
for all 
and
. A G-ring M equipped with an involution is called a G-ring M with involution.
Definition 2. An element x in a G-ring M with involution is said to be hermitian if 
and skew-hermi- tian if
. The sets of all hermitian and skew-hermitian elements of M will be denoted by 
and
, respectively.
Example 1. Let F be a field, and 
be a set of all diagonal matrices of order 2, with respect to the usual operation of addition and multiplication on matrices and the involution * on 
be defined by
with
, then we get 
and
.
Definition 3. An additive mapping 
is called a 
-derivation if 
for all 
and
.
To further clarify the idea of 
-derivation, we give the following example.
Example 2. Let R be a commutative ring with characteristic of R equal 2. Define 
and
, then M and 
are abelian groups under addition of matrices and M is a G-ring under multiplication of matrices.
Define a mapping 
by
.
To show that d is a 
-derivation, let
, 
, 
then
Now,
since
, this implies that 
then 
. Hence, d is a 
-derivation.
Definition 4. An additive mapping 
is called Jordan 
-derivation if 
for all 
and
.
Every 
-derivation is a Jordan 
-derivation, but the converse in general is not true as shown by the following example
Example 3. Let M be a G-ring with involution and let 
such that 
and 
for all 
and
, but 
for some 
such that
.
Define a mapping 
by 
for all 
and
. To show that d is a Jordan 
-derivation, we have on the one hand that
(3)
for all 
and
. On the other hand,
(4)
for all 
and
. If we compare Equations ((3), (4)), we get
then after reduction we get that d is a Jordan 
-derivation. Now to show that d is not a 
-derivation, we have on the one hand that
(5)
for all 
and
. On the other hand
(6)
for all 
and
. If we compare Equations ((5), (6)), we get
then after reduction we get that d is not a 
-derivation.
In this paper we will prove that if a non-zero Jordan 
-derivation d of a 2-torsion free semiprime G-ring M with involution satisfies 
for all 
and
, then
.
2. The Relation between Jordan G*-Derivation and 
on Semiprime G-Ring M with Involution
To prove our main results we need the following lemmas.
Lemma 1. Let M be a 2-torsion free semiprime G-ring with involution and 
be a Jordan 
- derivation which satisfies 
for all 
and
, then 
for all 
and
.
Proof. We have
(7)
for all 
and
. Putting 
for x in (7), we get
(8)
for all 
and
. Therefore,
for all 
and
. Setting 
in the above relation, we get
(9)
for all 
and
, because of
(10)
for all 
and
. According to (9) and (10), we get
(11)
for all 
and
. Then from relation (11)
for all 
and
. Since
, then
, and hence from the above relation
for all 
and
. Therefore,
for all 
and
. Then by using assumption (A), we obtain
(12)
for all 
and
. And
(13)
for all 
and
. Then from (13),
for all 
and
. Since
, then from the above relation
hence by using assumption (A), we obtain
(14)
for all 
and
. Therefore,
(15)
for all 
and
. Then from relation (15),
for all 
and
. By using relation (15) and assumption (A), we get
(16)
for all 
and
. Since M is 2-torsion free, we get
(17)
for all 
and
. Right multiplication of (17) by 
and using assumption (A), we get
(18)
for all 
and
. By semiprimeness of M, we have
(19)
for all 
and
. Left multiplication of (19) by z yields
(20)
for all 
and
. By semiprimeness of M again, we get 
for all 
and
. W
Lemma 2 Let M be a 2-torsion free semiprime G-ring with involution and 
be a Jordan 
- derivation which satisfies 
for all 
and
, then 
for all 
and
.
Proof. Putting 
for x in (7),
for all 
and
. By using Lemma (1), we get
(21)
for all 
and
. Replacing x by 
and y by 
yields
(22)
for all 
and
. Setting
, we get
(23)
for all 
and
. But
(24)
for all 
and
. So then from (23) and (24), we get
for all 
and
. Hence,
for all 
and
. Since
, then
, hence from the above relation,
(25)
for all 
and
. Therefore,
(26)
for all 
and
. Since
, we obtain
(27)
for all 
and
. Then from relation (27),
for all 
and
. By using relation (27) again,
(28)
for all 
and
. Since M is 2-torsion free, we get
(29)
for all 
and
. Right multiplication by z yields
(30)
for all 
and
. By semiprimeness of M, we therefore get 
for all 
and
. W
Remark 1 [17] . A G-ring M is called a simple G-ring if 
and its ideals are 0 and M.
Remark 2. Let M be a 2-torsion free simple G-ring with involution, then every 
can be uniquely represented in the form 
where 
and
.
Proof. Define
, 
, since 2M is an ideal of M and M is simple, it implies that
. So for every
, 
makes sense and so we can write
Now
hence
and
hence
Therefore
hence
. Let
, then 
and
, so 
and
. Therefore 
which implies that
, so
. Thus
. Hence 
implies that 
where 
and
. W
Theorem 1. Let M be a 2-torsion free semiprime G-ring with involution and 
be a Jordan 
- derivation which satisfies 
for all 
and 
and 
or 
for all
, 
and
, then 
for all 
and
.
Proof. Assume that 
for all
, 
and
. By using Lemma (1), we have
(31)
for all 
and
. For
, putting 
for h in (31) yields
for all 
and
. By using relation (31), we obtain
(32)
for all 
and
. Since 
for all
, then replace 
by 
in (32), to get
(33)
for all
, 
and
. By using Lemma (2), we have
(34)
for all 
and
. According to relations (33) and (34), we get
for all 
and
. By using Lemma (2), we get
(35)
for all 
and
. Then from relation (35),
for all
, 
and
. Therefore since
, we obtain
(36)
for all
, 
and
. Hence,
(37)
for all
, 
and
. Therefore,
for all
, 
and
. Then by using (37), we get
(38)
for all
, 
and
. Since M is 2-torsion free, we get
(39)
for all
, 
and
. Right multiplication of relation (39) by z yields
(40)
for all
, 
and
. By semiprimeness of M, we get
(41)
for all
, 
and
. Putting s for x and h for y in relation (21), we get
(42)
for all
, 
and
. Comparing relations (41) and (42), we get
(43)
for all
, 
and
. Since 
for all
, then from relation (43), we obtain
(44)
for all
, 
and
. Then
(45)
for all
, 
and
. Since 
for all
, 
and
, then from relation (45), we get
(46)
for all
, 
and
. Hence
(47)
for all
, 
and
. Then from relation (47),
for all
, 
and
. Then by using (47), we get
(48)
for all
, 
and
. Since M is 2-torsion free, we get
(49)
for all
, 
and
. Right multiplication of the relation (49) by z we get
(50)
for all
, 
and
. By semiprimeness of M, we get
(51)
for all
, 
and
. To prove
, since M is 2-torsion free, we only show
(52)
for all 
and
. By using Remark 2, we have for all 
and 
that 
for all 
and
. Therefore
for all
, 
and
. Hence
(53)
for all
, 
and
. By using Lemma (1), Lemma (2) and relation (41), (51), we get
Now assume 
for all
, 
and
. Since 
for all
, then we get
(54)
for all
, 
and
. Then from relation (54), we obtain
(55)
for all
, 
and
. Since 
for all
, 
and
, then from relation (55), we get
(56)
for all
, 
and
. Hence
(57)
for all
, 
and
. Then from relation (57),
for all
, 
and
. Then by using relation (57) we get
(58)
for all
, 
and
. Since M is 2-torsion free, we get
(59)
for all
, 
and
. Right multiplication of relation (59) by z yields
(60)
for all
, 
and
. By semiprimeness of M, we get
(61)
for all
, 
and
. Comparing relations (42) and (61), we get
(62)
for all
, 
and
. Since 
for all
, then from (62), we obtain
(63)
for all
, 
and
. Then,
(64)
for all
, 
and
. Since 
for all
, 
and
, then from relation (64), we get
(65)
for all
, 
and
. Hence
(66)
for all
, 
and
. Then from relation (66)
for all
, 
and
. Then by using (66) we get
(67)
for all
, 
and
. Since M is 2-torsion free, we get
(68)
for all
, 
and
. Right multiplication of relation (68) by z yields
(69)
for all
, 
and
. By semiprimeness of M, we get
(70)
for all
, 
and
. Therefore, by using Lemma (1), Lemma (2) and relation (61), (70), we get a similar result as the first assumption 
for all 
and
, and hence the proof of the theorem is complete. W
Acknowledgements
This work is supported by the School of Mathematical Sciences, Universiti Sains Malaysia under the Short-term Grant 304/PMATHS/6313171.
Cite this paper
Ali Kareem,Hajar Sulaiman,Abdul-Rahman Hameed Majeed, (2016) Jordan Γ*-Derivation on Semiprime Γ-Ring M with Involution. Advances in Linear Algebra & Matrix Theory,06,40-50. doi: 10.4236/alamt.2016.62006
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