Jordan Γ * -Derivation on Semiprime Γ-Ring M with Involution

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1 Advances in Linear Algebra & Matrix Theory, 206, 6, Published Online June 206 in SciRes. Jordan Γ -Derivation on Semiprime Γ-Ring M with Involution Ali Kareem, Hajar Sulaiman, Abdul-Rahman Hameed Majeed 2 School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia 2 Department of Mathematics, University of Baghdad, Baghdad, Iraq Received 2 March 206; accepted 30 May 206; published 2 June 206 Copyright 206 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). Abstract Let M be a 2-torsion free semiprime Γ-ring with involution satisfying the condition that abc = abc ( abc,, M and, Γ). In this paper, we will prove that if a non-zero Jordan Γ -derivation d on M satisfies d( x) x Z( M) Keywords, for all x M and Γ Γ-Ring M with Involution, Jordan Γ -Derivation, Commutative Γ-Ring, then d x, x.. Introduction The notion of Γ-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 Γ-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 Γ-rings. Nobusawa [] first introduced the notion of a Γ-ring and showed that Γ-rings are more general than rings. Barnes [2] slightly weakened the conditions in the definition of Γ-ring in the sense of Nobusawa. Barnes [2], Luh [3], Kyuno [4], Hoque and Pual [5]-[7], Ceven [8], Dey et al. [9] [0], Vukman [] and others obtained a large number of important basic properties of Γ-rings in various ways and developed more remarkable results of Γ-rings. We start with the following necessary introductory definitions. Let M and Γ be additive abelian groups. If there exists an additive mapping ( x,, y) ( xy) of M Γ M M which satisfies the conditions: ) x y M, x+ y z = xz+ yz, x + y = xy+ xy, x y+ z = xy+ xz, 2) How to cite this paper: Kareem, A., Sulaiman, H. and Majeed, A.-R.H. (206) Jordan Γ -Derivation on Semiprime Γ-Ring M with Involution. Advances in Linear Algebra & Matrix Theory, 6,

2 3) ( xy) z x( yz) ring need not be a ring. Let M be a Γ-ring. Then M is said to be prime if a M b ( 0) a or b and semiprime if aγmγ a = ( 0) with a M implies a commutative Γ-ring if xy = yx for all xy, M and Γ. Moreover, the set Z( M) { x M : xy yxfor all, y M} [ xy, ] = x y y x is known as the commutator of x and y with respect to, where, =, then M is called a Γ-ring [2]. Every ring M is a Γ-ring with M = Γ. However a Γ- Γ Γ = with ab, M, implies. Furthermore, M is said to be a = = Γ is called the center of the Γ-ring M. If M is a Γ-ring, then xy M and Γ. We make the following basic commutator identities: [ xyz, ] [ xz, ] y x[, ] y x[ yz, ] = + + () z [ xy, z] [ xy, ] z y[, ] z y[ xz, ] = + + (2) x for all xyz,, M and Γ., Now, we consider the following assumption: A... xyz = xyz, for all x, y, z M and, Γ. According to assumption (A), the above commutator identities reduce to [ xyz, ] = [ xz, ] y+ x[ yz, ] and [ xy, z] = [ xy, ] z+ y[ xz, ], which we will extensively used. During the past few decades, many authors have studied derivations in the context of prime and semiprime rings and Γ-rings with involution []-[4]. The notion of derivation and Jordan derivation on a Γ-ring were defined by [5]. Let M be Γ-ring. An additive mapping d : M M is called a derivation if d( xy) = d( x) y+ xd( y) for all xy, M and Γ. An additive mapping d : M M is called a Jordan derivation if d( xx) = d( x) x+ xd( x) for all x M and Γ. Definition [6]. An additive mapping ( xx) ( xx) on a Γ-ring M is called an involution if xy = y x and ( xx) = ( xx) for all xy, M and Γ. A Γ-ring M equipped with an involution is called a Γ-ring M with involution. Definition 2. An element x in a Γ-ring M with involution is said to be hermitian if x = x and skew-hermi- tian if x = x. The sets of all hermitian and skew-hermitian elements of M will be denoted by H( M ) and S( M ), respectively. Example. Let F be a field, and D2 ( F ) 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 D2 ( F ) be defined by a 0 a 0 a 0 a 0 = : a F with = : a, a F 0 a, then we get x = x and x = x. Definition 3. An additive mapping d : M M is called a Γ -derivation if d( xy) = d( x) y + xd( y) for all xy, M and Γ. To further clarify the idea of Γ -derivation, we give the following example. a b Example 2. Let R be a commutative ring with characteristic of R equal 2. Define M= : ab, R 0 and Γ= : R 0, then M and Γ are abelian groups under addition of matrices and M is a Γ-ring under multiplication of matrices. a b 0 b Define a mapping d : M M by d =. To show that d is a Γ -derivation, let a b c d c d x =, y = 0 c, y =, 0 c then a b 0 c d a b c d d( x y) = d d 0 0 c = 0 c ac ad + bc 0 ad + bc = d =. c 4

3 Now, 0 b 0 c d a b 0 0 d d( x) y + xd( y) = 0 0 c b c d a b 0 d = 0 c + 0 bc d 0 ad bc =, + = 0 since char ( R ) = 2, this implies that d( x) y xd( y) ( ) d x y = d x y + x d y. Hence, d is a ad + bc + = Γ -derivation. Definition 4. An additive mapping d : M M is called Jordan Γ -derivation if d( xx) = d( x) x + xd( x) for all x M and Γ. Every Γ -derivation is a Jordan Γ -derivation, but the converse in general is not true as shown by the following example Example 3. Let M be a Γ-ring with involution and let a M such that aγ a = ( 0) and xa x for all x M and Γ,, but xay 0 for some xy, M such that x y. Define a mapping d : M M by d( x) = xa+ ax for all x M and Γ. To show that d is a Jordan Γ -derivation, we have on the one hand that then ( ) = + = ( + ) + ( + ) d x x d x x x d x x a a x x x x a a x for all ax, M and Γ. On the other hand, = xax + ax x + xxa+ xax d x x = x x a+ a x x = x x a+ a x x (4) for all ax, M and Γ. If we compare Equations ((3), (4)), we get then after reduction we get that d is a Jordan have on the one hand that xax + ax x + xxa+ xax = xxa+ ax x Γ -derivation. Now to show that d is not a ( ) = + = ( + ) + ( + ) d x y d x y x d y x a a x y x y a a y = xay + ax y + xya+ xay for all axy,, M and Γ. On the other hand (3) Γ -derivation, we d x y = x y a+ a x y = x y a+ a y x (6) for all axy,, M and Γ. If we compare Equations ((5), (6)), we get xay + ax y + xya+ xay = xya+ ay x then after reduction we get that d is not a Γ -derivation. In this paper we will prove that if a non-zero Jordan Γ -derivation d of a 2-torsion free semiprime Γ-ring M with involution satisfies d ( x ), x Z ( M ) for all x M and Γ, then d( x), x 0 =. 2. The Relation between Jordan Γ -Derivation and d( x), x Z( M) Semiprime Γ-Ring M with Involution To prove our main results we need the following lemmas. Lemma. Let M be a 2-torsion free semiprime Γ-ring with involution and d : M M be a Jordan Γ - derivation which satisfies d ( x ), x Z ( M ) for all x M and Γ, then d( h), h 0 = for all on (5) 42

4 h H M and Γ. Proof. We have, A. Kareem et al. d x x Z M (7) for all x M and Γ. Putting x x for x in (7), we get for all x M and Γ. Therefore, ( ), d x x x x Z M (8) +, d x x x d x x x Z M for all x M and Γ. Setting x = h H( M) in the above relation, we get and Γ, because of +, d h h h d h h h Z M (9) 2, d h h+ h d h = h d h h d h (0) and Γ. According to (9) and (0), we get 2 hd h h, d h, h h Z M and Γ. Then from relation () 2 hd h, hh h, d h, hh Z M + 2 hd h, hh h, d h, hh Z M and Γ. Since d ( h ), h Z ( M ) the above relation (), then d( h), h, h h, and hence from [ ] 2 h d h, h h = 2 h d h, h h + h, h h d h Z M and Γ. Therefore, = + 2 h d h, hh 2 h h h, d h h, d h h = 2 hh hd, h + 2 h hd, h h = 2 hh hd, h + 2 h h hd, h Z M and Γ. Then by using assumption (A), we obtain and γ Γ., And and γ Γ.,, Then from (3), 4 hhγ hd, h Z M (2) 4 hhγ h, d h, d h ( hhγ h d( h) d( h) h h d( h) h d( h) + ) γ 4 ( hhd, ( h) γ hd, ( h) ) 4 ( h hd, ( h) hd, ( h) h) γ hd, ( h) 4 h hd, ( h) γ hd, ( h) 4 hd, ( h) hγ hd, ( h) 4,,,, = = + = + = 4 h hd, h γ hd, h + 4 hd, h hγ hd, h (3) 43

5 and γ Γ.,, Since hd, ( h) Z( M), then from the above relation hence by using assumption (A), we obtain + 4 h h, d h γ h, d h 4 h d h, h γ d h, h = 4 h d h, h γ d h, h + 4 h h, d h γ h, d h, and γ Γ.,, Therefore, 8 h γ hd, h hd, h (4) 8 hγ h, d h h, d h, d h and γ Γ.,, Then from relation (5), hγ ( h d( h) h d( h) ) d( h) h d( h) γ h d( h) h d( h) 8,,, + 8,,, = 0 and γ Γ.,, By using relation (5) and assumption (A), we get ( hd( h) hd( h) ) hd( h) (5) 8,, δ, (6) and γ Γ.,, Since M is 2-torsion free, we get we get ( hd( h) hd( h) ) hd( h),, δ, (7) and γ Γ.,, Right multiplication of (7) by z δ hd, ( h) and using assumption (A), ( hd( h) hd( h) ) z hd( h) hd( h),, δ δ,, (8) and γ Γ.,, By semiprimeness of M, we have hd( h) hd, h, (9) and γ Γ.,, Left multiplication of (9) by z yields and γ Γ.,, d h, h z d h, h (20) and γ Γ.,, By semiprimeness of M again, we get d( h), h Lemma 2 Let M be a 2-torsion free semiprime Γ-ring with involution and d : M M be a Jordan Γ - d x, x Z M for all x M and Γ, then d( s), s 0 = for all s S( M) and Γ. Proof. Putting x+ y for x in (7), derivation which satisfies for all h H( M) ( + ), + =, +, +, +, d x y x y d x x d x y d y x d y y Z M for all xy, M and Γ. By using Lemma (), we get ( d ( x ), y + d ( y ), x ) Z ( M ) for all xy, M and Γ. Replacing x by x x and y by (2) x yields ( d( xx), x ) + d x, xx Z( M) for all x M and Γ. Setting x = s S( M), we get and Γ. But ( d( ss), s ) + d s, ss Z( M) (22) (23) 44

6 ( ) = + = =, d s s d s s s d s s d s d s s s d s Z M (24) and Γ. So then from (23) and (24), we get and Γ. Hence, ( s, d( s), s ) + d s, s s Z( M) ( d ( s ), s, s + s s, d ( s ) ) Z ( M ) for all s S( M) and Γ. Since d ( s ), s Z ( M ) relation, and Γ. Therefore,, then d( s), s, s, hence from the above Z( M) s sd, s (25) = + = ssd, s s sd, s sd, s s 2 s sd, s Z M (26) for all s S( M) and Γ. Since sd, ( s) Z( M), we obtain and Γ. Then from relation (27), 2 s s, d s, d s ( s s d( s) d( s) s d( s) γ s d( s) ) 2,, +,, and Γ. By using relation (27) again, sd( s) and Γ. Since M is 2-torsion free, we get (27) 2 sd, s γ, (28) and Γ. Right multiplication by z yields d s, s γ d s, s (29) d s, s γzγ d s, s (30) Proof. Define = { = }, { } and Γ. By semiprimeness of M, we therefore get d( s), s s S M and Γ. Remark [7]. A Γ-ring M is called a simple Γ-ring if MΓM nd its ideals are nd M. Remark 2. Let M be a 2-torsion free simple Γ-ring with involution, then every x M can be uniquely represented in the form 2x = h+ s where h H( M) and s S( M). H M x M; x x S M = x M; x = x, since 2M is an ideal of M and M is simple, it implies that 2M Now hence and = M. So for every x M, x 2 makes sense and so we can write x+ x x x x = x+ x x+ x = ( x+ x ) = x + ( x ) = x + x = x+ x = x+ x 2 H M for all 45

7 hence Therefore x x x x = ( x x ) = x ( x ) = x x = x x = x x 2 S M x+ x x x x = + H M + S M 2 2 hence M = H( M) + S( M). Let x H( M) S( M), then x H( M) and x S( M), so x x = x. Therefore x = x which implies that 2x, so x. Thus H( M) S( M) 0 2M = M = H( M) + S( M) implies that 2x = h+ s where h H( M) and s S( M) = x and =. Hence. Theorem. Let M be a 2-torsion free semiprime Γ-ring with involution and d : M M be a Jordan Γ - derivation which satisfies d, Z for all x M and Γ and, or d, Z and Γ, then [ d ( x), x] for all x M and Γ. Proof. Assume that d, Z and Γ. By using Lemma (), we have d( h), h (3) and Γ. For h, h2 H( M), putting h+ h2 for h in (3) yields d h + h, h + h = d h, h + d h, h + d h, h + d h, h and Γ. By using relation (3), we obtain d h, h + d h, h (32) 2 2 for all h, h H( M) and Γ. Since s s H( M) to get 2 ( ), then replace h 2 by s s in (32), d h, s s + d s s, h (33) for all h H( M) and Γ. By using Lemma (2), we have d( ss) d( s) s sd( s) sd( s) d( s) s s d( s) = + = =, (34) and Γ. According to relations (33) and (34), we get d h, s s + s, d s, h = d h, s s d s, s, h and Γ. By using Lemma (2), we get and Γ. Then from relation (35), d h, s s 0 = (35) s d h, s + d h, s s and Γ. Therefore since d ( h ) s Z ( M ) and Γ. Hence, s d h s,, we obtain 2, (36) 2,, d h s d h s γ (37) 46

8 and γ Γ.,, Therefore, γ γ 2 d h, s d h, s 2 s d h, d h, s = + d h, s d h, s γ and γ Γ.,, Then by using (37), we get A. Kareem et al. 2 d h, s γ d h, s (38) and γ Γ., Since M is 2-torsion free, we get d h, s γ d h, s (39) and γ Γ., Right multiplication of relation (39) by z yields d h, s γzγ d h, s (40) and γ Γ., By semiprimeness of M, we get d h, s (4) and Γ. Putting s for x and h for y in relation (2), we get, +, d s h d h s Z M (42) and Γ. Comparing relations (4) and (42), we get, d s h Z M (43) and Γ. Since hγ h H( M) we obtain, γ, then from relation (43), d s h h Z M (44) and Γ. Then d( s), hγh = hγ d( s), h + d( s), h γh Z( M) (45) and Γ. Since d ( s ), h Z ( M ) Γ, then from relation (45), we get and γ Γ., Hence and 2 hγ d s, h Z M (46) γ 2 d s, h d s, h and γ Γ.,, Then from relation (47), ( hγ d( s) d( s) h d( s) h γ d( s) h ) 2,, +,, and γ Γ.,, Then by using (47), we get (47) 2 d s, h γ d s, h (48) and γ Γ., Since M is 2-torsion free, we get d s, h γ d s, h (49) and γ Γ., Right multiplication of the relation (49) by z we get d s, h γzγ d s, h (50) and γ Γ., By semiprimeness of M, we get 47

9 d( s), h (5) and Γ. To prove d( x), x, since M is 2-torsion free, we only show 4 d x, x (52) for all x M and Γ. By using Remark 2, we have for all x M and Γ that 2x = s+ h for all s S M h H M. Therefore and h 4 d x, x = d 2 x,2 x = d s+ h, s+ h and Γ. Hence 4 d x, x = d s, s + d h, s + d s, h + d h, h (53) and Γ. By using Lemma (), Lemma (2) and relation (4), (5), we get d( x), x. Now assume d ( s ), h Z ( M ) and Γ. Since h h H( M) H( M), then we get d( s), hγ h Z( M) γ for all (54) and γ Γ., Then from relation (54), we obtain, γ = γ, +, γ d s h h h d s h d s h h Z M (55) and Γ. Since d ( s ), h Z ( M ) Γ, then from relation (55), we get and γ Γ., Hence and 2 hγ d s, h Z M (56) γ 2 d s, h d s, h and γ Γ.,, Then from relation (57), ( hγ d( s) d( s) h d( s) h γ d( s) h ) 2,, +,, and γ Γ.,, Then by using relation (57) we get (57) 2 d s, h γ d s, h (58) and γ Γ., Since M is 2-torsion free, we get d s, h γ d s, h (59) and γ Γ., Right multiplication of relation (59) by z yields d s, h γzγ d s, h (60) and γ Γ., By semiprimeness of M, we get d( s), h (6) and Γ. Comparing relations (42) and (6), we get, d h s Z M (62) and Γ. Since sγ s H( M) for all s S( M), then from (62), we obtain 48

10 , γ d h s s Z M (63) and Γ. Then, d( h), sγs = sγ d( h), s + d( h), s γs Z( M) (64) and Γ. Since d ( h ), s Z ( M ) Γ, then from relation (64), we get and γ Γ., Hence and 2 sγ d h, s Z M (65) γ 2 d h, s d h, s and γ Γ.,, Then from relation (66) ( sγ d( h) d( h) s d( h) s γ d( h) s ) 2,, +,, and γ Γ.,, Then by using (66) we get (66) 2 d h, s γ d h, s (67) and γ Γ., Since M is 2-torsion free, we get d h, s γ d h, s (68) and γ Γ., Right multiplication of relation (68) by z yields d h, s γzγ d h, s (69) and γ Γ., By semiprimeness of M, we get d( h), s (70) and Γ. Therefore, by using Lemma (), Lemma (2) and relation (6), (70), we get a similar result as the first assumption d( x), x for all x M and Γ, and hence the proof of the theorem is complete. Acknowledgements This work is supported by the School of Mathematical Sciences, Universiti Sains Malaysia under the Short-term Grant 304/PMATHS/6337. References [] Nobusawa, N. (964) On a Generalization of the Ring Theory. Osaka Journal of Mathematics,, [2] Barnes, W.E. (966) On the Γ-Rings of Nobusawa. Pacific Journal of Mathematics, 8, [3] Luh, J. (969) On the Theory of Simple Gamma-Rings. The Michigan Mathematical Journal, 6, [4] Kyuno, S. (978) On Prime Gamma Rings. Pacific Journal of Mathematics, 75, [5] Hoque, M.F. and Paul, A.C. (202) An Equation Related to Centralizers in Semiprime Gamma Rings. Annals of Pure and Applied Mathematics,, [6] Hoque, M.F. and Paul, A.C. (204) Generalized Derivations on Semiprime Gamma Rings with Involution. Palestine Journal of Mathematics, 3, [7] Hoque, M.F. and Paul, A.C. (203) Prime Gamma Rings with Centralizing and Commuting Generalized Derivations. International Journal of Algebra, 7,

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