Research Journal of Applied Sciences, Engineering and Technology 6(22): 4129-4137, 2013 ISSN: 2040-7459; e-ISSN: 2040-7467 © Maxwell Scientific Organization, 2013 Submitted: November 19, 2012 Accepted: May 07, 2013 Published: December 05, 2013 The θ-Centralizers of Semiprime Gamma Rings 1 1 M.F. Hoque, 1H.O. Roshid and 2A.C. Paul Department of Mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh 2 Department of Mathematics, Rajshahi University, Rajshahi-6205, Bangladesh Abstract: Let M be a 2-torsion free semiprime Γ-ring satisfying a certain assumption and θ be an endomorphism of M. Let T : M  M be an additive mapping such that T ( xy x )   ( x )T ( y )  ( x ) holds for all x, y M and  ,    . Then we prove that T is a θ-centralizer. We also show that T is a θ-centralizer if M contains a multiplicative identity 1. 2010 Mathematics Subject Classification, Primary 16N60, Secondary 16W25, 16U80. Keywords: Centralizer, θ-centralizer, Jordan centralizer, Jordan θ-centralizer, left centralizer, left θ-centralizer, semiprime Γ-ring, INTRODUCTION The notion of a Γ -ring was first introduced as an extensive generalization of the concept of a classical ring. From its first appearance, the extensions and generalizations of various important results in the theory of classical rings to the theory of Γ-rings have been attracted a wider attentions as an emerging field of research to the modern algebraists to enrich the world of algebra. Many prominent mathematicians have worked out on this interesting area of research to determine many basic properties of Γ-rings and have extended numerous remarkable results in this context in the last few decades. All over the world, many researchers are recently engaged to execute more productive and creative results of Γ-rings. We begin with the definition. Let M and Γ be additive abelian groups. If there exists a mapping ( x, , y)  xy of MM M , which satisfies the conditions:  ( xy )  M ( x  y )z  xz  yz , x(   ) z  xz  xz ,  ( x α y ) β z = x α ( y β z ) for  x ( y  z )  xy  xz and  ,    , then M is called a all x, y , z  M Let M be a Γ -ring. Then an additive subgroup U of M is called a left (right) ideal of M if MU  U (UM  U ) . If U is a left and a right ideal, then we say U is an ideal of M . Suppose again that M is a Γ.-ring. Then M is said to be a 2-torsion free if 2x  0 implies x  0 for all x  M . An ideal P1 of a Γ -ring M is said to be a prime if for any ideals A and B of M, AB  P1 implies A  P1 or B  P1 . An ideal P2 of a Γ-ring M is said to be semiprime if for any ideal U of M, UU  P2 implies U  P2 . A Γ-ring M is said to be prime if a  M  b  ( 0 ) with, a, b  M implies a  0 or b  0 and semiprime if a  0. implies aMa  (0) with a  M Furthermore, M is said to be commutative Γ-ring if xy  yx for all x, y  M and    . Moreover, the set Z(M) {x M : xy  yx for all y  M } is called the centre of the Γ-ring M. If M is a  -ring,  , where x, y  M and commutator identities: Every ring M is a Γ-ring with M = Γ. However a Γ -ring need not be a ring. Gamma rings, more general than rings, were introduced by Nobusawa (1964). Bernes (1966) weakened slightly the conditions in the definition of Γ-ring in the sense of Nobusawa (1964). and then [ x, y ]  x y  y x is    . We make the basic known as the commutator of x and  -ring.   y with respect to [ xy, z]  [ x, z] y  x[ ,  ] z y  x[ y, z] and [ x , y  z ]   [ x , y ]   z  y[ ,  ] x z  y  [ x , z ]  for all x , y , z  M and  ,    . We consider the following assumption: Corresponding Author: M.F. Hoque, Department of Mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh 4129 Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013  x y z  x y z ………….. x, y , z  M and  ,    . (A) for all According to the assumption (A), the above two identities reduce to [ xy, z ]   [ x, z ]  y  x [ y, z ]  and [ x, yz ]   [ x, y ]   z  y [ x, z ]  ,which we extensively used. An additive mapping T : M  M is a left (right) centralizer if T ( xy )  T ( x)y[T ( xy )  xT ( y )] holds for all x, y, z  M and  ,    . A centralizer is an additive mapping which is both a left and a right centralizer. For any fixed and Г,, the mapping is a left centralizer and is a right centralizer. We shall restrict our attention on left centralizer, since all results of right centralizers are the same as left centralizers. An additive mapping ∶ ⟶ is called a derivation if holds for all , Г and is called a Jordan derivation if for all and Г. An additive mapping ∶ ⟶ is Jordan left (right) centralizer if for all and Г. Every left centralizer is a Jordan left centralizer but the converse is not in general true. An additive mappings ∶ ⟶ is called a Jordan centralizer if , for all , and Г . Every centralizer is a Jordan centralizer but Jordan centralizer is not in general a centralizer. Bernes (1966), Luh (1969) and Kyuno (1978) studied the structure of Г -rings and obtained various generalizations of corresponding parts in ring theory. Zalar (1991) worked on centralizers of semiprime rings and proved that Jordan centralizers and centralizers of this rings coincide. Vukman (1997, 1999, 2001) developed some remarkable results using centralizers on prime and semiprime rings. Ceven (2002) worked on Jordan left derivations on completely prime Г -rings. He investigated the existence of a nonzero Jordan left derivation on a completely prime Г-ring that makes the Г-ring commutative with an assumption. With the same assumption, he showed that every Jordan left derivation on a completely prime Г-ring is a left derivation on it. The commutativity condition of a prime ring has been studied by Mayne (1984) by means of a Lie ideal of R and with a nontrivial automorphism or derivation. Ullah and Chaudhary (2010) proved that if T is an additive mapping of a 2-torsion free semiprime ring with involution * satisfying T (xx* )  T (x) (x* )   (x* )T (x) , then t is a -centralizer. Hoque and Paul (2011) have proved that every Jordan centralizer of a 2-torsion free semiprime Г -ring satisfying a centain assumption is a centralizer. Hoque and Paul (2012) have proved that if T is an additive mapping on a 2-torsion free semiprimeГ -ring M with a certain assumption such that , for all , and , Г , then is a centralizer. Ullah and Chaudhary (2012), have proved that every Jordan -centralizer of a 2-torsion free semiprime Г -ring is a -centralizer. Hoque et al. (2012) proved that T is a -centralizer by using a relation 2T (aba)  T (a)(b)(a)  (a)(b)T (a) , where T is an additive mapping on a 2-torsion free semiprimeГ -ringM. Our research works inspired by the works of Hoque et al. (2012) in Г -rings with -centralizers. Here we prove that if is a 2-torsion free semiprime Г-ring satisfying the assumption (A) and if ∶ ⟶ is an additive mapping such that: (1) For all , , , Г and an endomorphism on , then is -centralizer. Also we show that is a centralizer if contains a multiplicative identity 1. THE -CENTRALIZER OF SEMIPRIME GAMMA RINGS In this section, we have given the following definitions: Let be a 2-torsion free semiprime Г -ring and let be an endomorphism of . An additive mapping ∶ ⟶ is a left (right) -centralizer if holds for all , and Г. If t is a left and a right -centralizer, then it is natural to call a -centralizer. Let be a Г-ring and let and Г be fixed element. Let ∶ ⟶ be an endomorphism. Define a mapping ∶ ⟶ by . Then it is clear that is a left -centralizer. If is defined, then is a right -centralizer. An additive mapping ∶ ⟶ is a Jordan left (right) -centralizer if = holds for all and Г. It is obvious that every left -centralizer is a Jordan left -centralizer but in general Jordan left -centralizer is not a left -centralizer. Example 2.1 Let be a Г-ring and let ∶ ⟶ be a left -centralizer, where ∶ ⟶ is an endomorphism. Define , ∶ and Γ , ∶ Γ . The addition and multiplication on are defined as follows: , , , and , , , , . 4130 Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013 Under these addition and multiplication, it is clear that is a Γ -ring. , Define , , and , . Then T is an additive mapping on and is an endomorphism on . Now, let , , , ,then we have: , , , , , , , . , , , , , , . Under these addition and multiplication is a Γ -ring.Define , , and , , . Then is additive mapping and is an endomorphism on . Let , , , , , . Then we have: , , , , , , , , , , , , , Therefore, T is a Jordan left -centralizer which is not a left -centralizer. Let be a Γ-ring and let be an endomorphism on . An additive mapping ∶ ⟶ is called a Jordan -centralizer if , for all , and Г. It is clear that every -centralizer is a Jordan -centralizer but the converse is not in general a -centralizer. An additive mapping ∶ ⟶ is a , derivation if holds for all , and Г and is called a Jordan , -derivation if , holds for all and Г .Now we begin with two examples which are ensure that a centralizer and a Jordan -centralizer exist in Γ-ring. Therefore, is a Jordan -centralizer on which is not a -centralizer. For proving our main results, we need the following Lemmas: Example 2: Let be a Γ-ring satisfying the assumption (A) and let be a fixed element of such that , the centre of . Define a mapping ∶ ⟶ by , where ∶ ⟶ is an endomorphism and Г is a fixed element.Then for all , and Г , we have: Lemma 2: Let be a -torsion free semiprimeΓ –ring satisfying the assumption and let ∶ → be an additive mapping. Suppose that holds for all , ∈ and , ∈ Γ . Then: , Also, Therefore, is a . is a left and right . , , , , , Lemma 1: (7, Lemma 1) Suppose is a semiprime Γring satisfying the assumption . Suppose that the relation holds for all ∈ , some , , ∈ and , ∈ Γ. Then is satisfied for all ∈ and , ∈ Γ. , , , , Proof: We prove (i) , , , T x , x  , x  , , , . 0 For linearization, we put relation(1),we obtain: centralizer, so Example 3: Let be a Γ-ring and let ∶ → be a -centralizer, where ∶ → is an endomorphism. , : ∈ and Γ , : ∈Γ . Define as follows: Define addition and multiplication on , , , and , , xz (2) for x in T  x  y  z  z  y  x     x  T  y   z     z  T  y   x  (3) Replacing y for x and z for y in (3), we have: T  x  x  y  y  x  x     x  T  x   x     y  T  x   x  4131 (4) Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013 For z   x  x in relation (3) reduces to:   , T x y  x   x   x  x y x 2 2    x  T  y  x     x     x     x  T  y   x  (5) Putting y  xyx in (4), we obtain: 2 2     2  2  , The substitution xx y  yxx for y in the relation (1) gives:   T x  xyx   x  xyx    x T  xxy  yxx   x   2 which gives because of (4):  T  x  x y x  x y  x  x 2    x  T  x   y   x     x   y  T  x   x   x  (7) 2 2 , (11) one obtains: , Since is semiprime, so , , Now, we prove the relation ∶ , follows, i.e., , (12) The linearization of (2) gives: ,  y  x    x    y  T  x ,  x   x   0 gives: , , Combining (6) and (7), we arrive at: T  x  T  x ,   x   from (10) , , Subtracting (6) 2 , Right multiplication of 9 by T  x  xyx  x y  x    xT xx yx   x yxT x x 2 , (8) Putting , , , ,  x   y  T x   x  x     x   y  x  T  x  x    x  T x ,  x    y  x   0 , , , in the above relation, we have: , Using Lemma 2 in the above relation , we have: , , , , , , , , , , , , , , Adding the above two relations, we have: , , , , , Let , , , , , , , , , , Since M is 2-torsion free semiprime Г-ring, so, we have: , , , , , , (13) Putting for y in (13) and using (1), (2), (13) and assumption (A), we have: , , (9) 9 , 4132 , , , , , , Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013 , , , , , , , , , , , , , [T x , x  x  x ]  T x , x   x   x   y  T x , x   0 which reduces to:  x  T x , x   x    x  x   T x ,  x    y  T x ,  x  , , , T x , x   x  makes it possible to write instead of   x  T  x ,   x  , which means that   x   x  T  x ,   x  can be Relation   x  T  x ,   x   x   y  T  x ,   y   0 Therefore, we have: \ , (2) replaced by   x  T  x ,   x    x  in the above relation. Thus we have: , , T x ,θ x θ y \θ x γθ x θ x γθ x θ y T x , θ x θ x γθ y θ x T x , θ x _ T x , θ x _ θ x θ y γθ x . , Right multiplication of the above relation by and substitution for gives finally, , , , , Hence For all x,y ∈ , ∈ , which reduces because of (3) and (8) to: Left multiplication of the above relation by xβ gives: , by One can replace in the above relation according to (15), θ(x)β[T(x), θ(x)]α βθ(y)βθ(x) by   x   y  T  x ,   x    x  which gives:   x   y  T x , x   x  x  , of ∈ for for , we  ∈ have (17) in (8), gives Right multiplication of (18) by in (25), we have: θ , (15) M, (18) , (19) , (20) Subtracting (20) from (19), we have: The substitution T(x)αθ(y) for y in (14), we have:  x  T x   y  T x ,   x   x  x     x   x   x T  x   y  T  x ,   x   0 (16) ⇒ ⇒ Subtracting (16) from (15), we obtain: T x , x    y  T x , x   x  x   T  x ,   x   x   x    y  T  x ,   x  , First, we putting because of (12): (14) T  x   x   y  T  x ,   x    x   x   T  x   x   x   x   y  T  x , x   0 , The substitution Left multiplication of the above relation by T(x)α gives: semiprimeness . Next we prove the relation (iii):  x  T x ,  x    y  x  x   y  T x , x   0    x   x   x   y  T  x ,   x   0 0 ⇒ 0 From the above relation and Lemma-2.1, it follows that: According , gives: 4133 , , , , , , to , , , (2), by one can , replaced which Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013 , , ∈ , , , ∈   ( x ) (T ( xy  yx )  T ( y ) ( x ) ,   ( x )T ( y ))  ( x)  0. Hence by semiprimenessof M,  ( x) [T ( x), ( x)]  0 , We define: Finally, we prove the equation (u) : G ( ( x ),  ( y ))  T ( xy  yx )  T ( y ) ( x )   ( x )T ( y ). x, y  M ,  ,   . T ( x ),  ( x) = 0 Then it is clear that: (21)  ( x )G ( ( x ),  ( y ))  ( x)  0 and G ( ( x ),  ( y ))  G ( ( y ),  ( x )). From (2) and (17), it follows that: [T ( x),  ( x)]  ( x) = 0, x  M ,  ,   . Replacing x for y and using (22), we have:  ( y )G ( ( x), ( y )) ( y )  0. We can also easily prove the following results :  The linearization of the above relation gives (see how relation (13) was obtained from (2)): [T ( x ),  ( x )]  ( y )  [T ( x ),  ( y )] (1) G ( ( x)   ( z ), ( y ))  G ( ( x), ( y ))  G ( ( z), ( y)) (2) G ( (x), ( y)   (z))  G ( (x), ( y))  G ( ( x), ( z ))  ( x )  [T ( y ),  ( x )]  ( x )  0 (3) G   ( ( x), ( y ))  G ( ( x), ( y ))  G  ( ( x), ( y )) (4) G ( ( x), ( y))  G ( ( x), ( y)) (5) G ( ( x), ( y ))  G ( ( x), ( y )) . Right multiplication of the above relation by  [T ( x), x] gives because of (17): [T ( x), ( x)]  ( y )  [T ( x), ( x)]  0 which implies [T ( x ),  ( x )]  0. Lemma 3: Let M be  - ring satisfying the assumption (A) and let T : M  M be an additive mapping such that T ( xyx)   ( x)T ( y )  ( x) holds for all x, y, M and  ,   . Then:  ( x) (T ( xy  yx)  T ( y) ( x)   ( x)T ( y)) ( x)  0 Proof: The substitution gives: xy  yx (22) for Lemma 4: Let M be a 2-torsion free semiprime Γring satisfying the assumption (A) and let T : M  M be an additive mapping. Suppose that T ( xyx)   ( x)T ( y)  ( x) holds for all x, y  M and  ,    . Then: (a) [G ( ( x),  ( y )),  ( x)]  0 (b) Gα(θ(x),θ)) = 0 Proof: First we prove the relation (a): [G ( ( x), ( y)), ( x)]  0 y in (1) T ( xxyx  xyxx)   ( x)T ( xy  yx)  ( x) The linearization of (21) gives: [T ( x),  ( y )]  [T ( y ),  ( x)]  0 (23) On the other hand, we obtain by putting z  xx in (3), we have: T ( xxy x  xy xx )   ( x)T ( y ) ( x )   ( x)   ( x) ( x)T ( y )  ( x) T ( xxyx  xyxx)   ( x)T ( y )  ( x)  ( x)   ( x) ( x)T ( y )  ( x) (24) (25) (26) Putting xy  yx for y in the above relation and using (21), we obtain: [T ( x), ( x) ( y)   ( y) ( x)]  [T ( xy  yx), ( x)]  0   xT  x,  y  T x,  y  x  T xy  yx,  x  0  T xy  yx, x   x T x,  y  T x,  y x  0 By comparing (23) and ( 24), we have: 4134 Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013 According to (26) one can replace T  x ,   y  by  T  y , x  which gives G  ( ( x ),  ( y ))  0 i.e., T ( xy  yx)  T ( y) ( x)   ( x)T ( y) in the above relation. We have therefore: T xy  yx , x    x  T  y , x   T  y ,  x   x   0 i.e., G   x ,   y ,   x   0 Hence the proof is complete. Theorem 1: Let M be a 2-torsion free semiprime Γ-ring satisfying the assumption (A) and let T : M  M be an additive mapping. Suppose that T(xαxβx) = θ(x)αT(x)βθ(x) holds for all x  M and  ,    . Then T is a  -centralizer. The proof is therefore complete. Finally, we prove the relation (b): G   x ,   y   0 Proof: In particular for reduces to: (27) Combining the above relation with (21), we arrive at: Right multiplication of the above relation by G   x ,  y   x  gives because of (22):   x  G   x ,   y   z  G    x ,   y   x   0 2T ( xx)  2T ( x) ( x), xM ,   2T ( xx)  2 ( x)T ( x), x  M ,   . And Since (28) Relation (25) makes it possible to replace in (28),  x G  x ,  y  by G   x ,  y   x  . Thus we have: G  x ,   y  x  z  G   x ,   y  x   0 (29) M is 2-torsion free, so we have: T ( xx)  T ( x) ( x), T  xx    xT x, x  M,   x  M ,   By theorem 3.5 in Ullah and Chaudhary (2012), it follows that T is a left and also right θ–centralizer which completes the proof of the theorem. Putting y = x in relation (1), we obtain: T  x  x  x     x  T  x   x , x  M ,  ,   T (33) Therefore by semiprimeness of M: G   x ,   y   x   0 the relation (32) 2T ( x x )  T ( x ) ( x )   ( x ) T ( x ) From (22) one obtains (see how (13) was obtained from (2)), we have:  x G  x ,   y  z    x G  z ,   y  x    z G  x ,   y   0 y  x, The question arises whether in a2-torsion free semi-prime Γ–ring the above relation implies that T is a θ-centralizer. Unfortunately we were unable to answer affirmative if M has an identity element. (30) Of course, we have also:  ( x )G ( ( x),  ( y ))  0 Theorem 2: Let M be a 2-torsion free semi-prime Γ– ring with identity element 1 satisfying the assumption (A) and let T : M  M be an additive mapping. Suppose that T  xxx    xT  x x holds for The linearization of (30) with respect to x gives: G ( ( x),  ( y)) ( z)  G ( ( z),  ( y)) ( x)  0 all x  M and  ,    . Then T is a Right multiplication of the above relation by G ( ( x),  ( y)) gives because of (31):  -centralizer. Proof: Putting x+1 for x in relation (33), one obtains after some calculations: G ( ( x),  ( y )) ( z )G ( ( x),  ( y ))  0 4135 3T  xx   2T  x   T  x   x   x T  x     x a  x   a  x     x a Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013 where a stands for T(1). Putting –x for x in the relation above and comparing the relation so obtain with the above relation we have: 6T xx   2T  x   x   2  x T x   2 x a x  (34) and and θ be an endomorphism of M. Suppose there exists an additive mapping T : M  M such that m n m n holds for all T (( x ) ( x ) x)   ( x ) T ( x) (  x ) x  M ,  ,    , where m  1, n  1 are some integers. Thus T is a θ-centralizer. CONCLUSION 2T  x   a x    x  (35) We shall prove that a  Z M  .According to (35) one can replace 2T  x  on the right side of (41) by a x     x  and 6T xx  on the left side by 3a x  x   3 x  x a , which gives after some calculation: a x  x     x   x a  2  x a  x   0 REFERENCES The above relation can be written in the form: [[ a , ( x )] ,  ( x )]   0; x  M ,  ,    (36) The linearization of the above relation gives: [[ a,  ( x )] , ( y )]   [[ a, ( y )] , ( x )]   0 In this study, we have given some examples which have shown that θ-centralizer exists in Γ-rings. We proved that if T is an additive mapping on a 2-torsion free semiprime Γ-ring M satisfying the assumption (A) such that T ( x  y  x )   ( x ) T ( y )  ( x ) for all x, y  M ,  ,    and θ an endomorphism on M,then T is a θ-centralizer. We have also showed that T is a θcentralizer if M contains a multiplicative identity 1. (37) Putting y  x y in (37), we obtain because of (36) and (37): 0  [[a,  ( x)] ,  ( x) ( y)]  [[ ,  ( x) ( y)] ,  ( x)]  [[a, ( x)] , ( x)]  ( y)]   ( x) )[[a, ( x)], ( y)]  [[a, ( x)]  ( y)   ( x)[a, ( y)] , ( x)]   ( x)[[a, ( x)] , ( y)]  [[a, ( x)]   ( y), ( x)]  [ ( x)[a, ( y)] , ( x)]   ( x) [[a, ( x)] , ( y)]  [[a, ( x)  ( x)]  ( y)  [a, ( x)]  [ ( y), ( x)]  [ ( x) [a, ( y)] , ( x)]  [a, ( x)]  [ ( y), ( x)] The substitution  ( y ) a for  ( y) in the above relation gives: [ a, ( x )]  ( y )  [a, ( x)]  0 Hence it follows a  Z (M ) , which reduces (35) to the form T ( x)  a ( x), x  M ,  . The proof of the theorem is complete. We conclude with the following conjecture: let M be a semiprime Γ-ring with suitable torsion restrictions Bernes, W.E., 1966. On the Γ-rings of nobusawa. Pacific J. Math., 18: 411-422. Ceven, Y., 2002. 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