On Right α-centralizers and Commutativity of Prime Rings

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1 On Right α-centralizers and Commutativity of Prime Rings Amira A. Abduljaleel*, Abdulrahman H. Majeed Department of Mathematics, College of Science, Baghdad University, Baghdad, Iraq Abstract: Let R be a prime ring, U be a lie ideal of R, and α be an endomorphism of R. An additive mapping is called a right α-centralizer if holds for all. In the present paper, we shall investigate the commutativity of prime rings admitting right α-centralizer satisfying any one of the following conditions: (i) ([ ]) [ ], (ii), (iii) for all. Keywords: Prime Ring, Right (Left) α- Centralizer. ISSN: α حول التمرك ازت اليمينية واالبدالية للحلقات االولية * amaaa605@yahoo.com 134 الخالصة: اميرة عامر لتكن R حلقة اولية, U مثالي لي في R عبد الجليل *, عبد الرحمن حميد مجيد قسم الرياضيات, كلية العلوم, جامعة بغداد, بغداد, الع ارق و α هي تشاكل على R. الدالة الجمعية تتحقق لكل تسمى. في بحثنا هذا سوف نتحقق من االبدالية, تمركز -α ايمن اذا ) ( للحلقات االولية المعرف عليها لكل تمركز α- ايمن اذا حققت احد هذة الشروط: )1( ] [ ([ ]). (3(, 1. Introduction: Throughout the present paper, R will represent an associative ring with center Z(R). A ring R is called n-torsion free, where is an integer, if whenever with, then. For any, the symbol [ ] will denote the commutator and the symbol will denote the anticommutator. Recall that a ring R is called prime if for any, implies that either or. A ring R is called semiprime if for any, implies. A prime ring is clearly a semiprime. If S is a subset of R, then we can define the left (resp. right) annihilator of S as (resp. for all ). An additive subgroup U of R is said to be a lie ideal of R if [ ] for all and. In [1], Zalar, introduce the concept of centralizer in a ring, an additive mapping is called right (resp. left) centralize if (resp. ) holds for all, and f is called a centralizer if it is a right as well as a left centralizer. In [2], Bresar, introduce the concept of generalized derivation, an additive mapping is called a generalized derivation if there exist a derivation such that holds for all, and d is called the associated derivation of f. Obviously, generalized derivation with covers the concept of left centralizer. In [3], Albas, introduce the notion of α-centralizer of R, an additive mapping is called a right (resp. left) α- centralizer if (resp. ) holds for all, whrer α is an endomorphism of R. If f is both right as well as left α-centralizer, then it is natural to call f an α- centralizer. It is clear that for an additive mapping associated with a homomorphism, if and for a fixed element and for all, then is a left α-centralizer and is a right α-centralizer. (2(

2 Clearly every centralizer is a special case of an α-centralizer with, the identity map on R. In [4], Rehman, discussed the commutativity of prime rings admitting a generalized derivation satisfying many conditions on a nonzero lie ideal. In [5], Ashraf and Ali, investigate the commutativity of prime rings satisfying certain identities involving left multiplier on a nonzero ideal. In [6], Ali and Huang, get many sufficient conditions of commutativity on left α-multipliers when R is semiprime ring and U denoted a nonzero ideal of R. In the present paper, we extend the above mentioned results for prime ring admitting a right α- centralizer where α is an endomorphism of R and U will always denote a lie ideal of R. 2. Some Preliminaries: In the following remark, we shall make extensive use of the basic commutator and anti- commutator identities. Remark 1: Let R a ring and, then (i) [ ] [ ] [ ], (ii) [ ] [ ] [ ], (iii) [ ] [ ], (iv) [ ] [ ], (v), The following lemmas are well-known results which will be used to prove our results in the next section. Lemma 1:[7] If U is a lie ideal of R such that for all, then for all. Lemma 2:[8] If is a lie ideal of R and if, then or. Lemma 3:[4] Let R be a 2-torsion free prime ring and U be a nonzero lie ideal of R. If U is a commutative lie ideal of R, then 3. The Main Results: In this section we prove the following results. Theorem 1: If ([ ]) [ ] for all, then. ([ ]) [ ] for all By using Lemma 1, replacing y by 2xy in, we get, ([ ]) [ ] for all ( [ ] [ ] ) [ ] [ ] ( [ ]) [ ] α ([ ]) [ ] [ ] [ ] ( )[ ] for all By using Lemma 1, replacing y by 2yz in (3), we get, ( )[ ] for all (4) ( ) [ ] ( )[ ] Using (3), we get, ( ) [ ] (5) 135

3 ( ) [ ] By using Lemma 2, we have, α [ ] Since α is not the identity map on U, then [ ] for all (6) U is commutative and hence, by Lemma 3. Theorem 2: If ([ ]) [ ] for all, then. ([ ]) [ ] By using Lemma 1, replacing y by 2xy in, we get, ([ ]) [ ] ( [ ] [ ] ) ( [ ] [ ] ) ( [ ]) ( [ ]) α ([ ]) ( [ ]) α( [ ]) ( [ ]) ( [ ]) ( [ ]) we get, [ ] [ ] ( )[ ] for all (3) By the same method in Theorem 1, we complete the proof. Theorem 3: If for all, then. By using Lemma 1, replacing x by 2yx in, we get, ( ) Using the fact that char R 2, we get, ( ) By Remark 1, this can be rewritten as, ( ) α α ( ) (3) By using Lemma 1, replacing x by 2xz in (3), we get, ( )( ) (4) 136

4 ( ) [ ] ( ) Using (3), we get, ( ) [ ] (5) ( ) [ ] By using Lemma 2, we have, α [ ] Since α is not the identity map on U, then α [ ] for all (6) U is commutative and hence, by Lemma 3. Theorem 4: If for all, then. By using Lemma 1, replacing x by 2yx in, we get, ( ) Using the fact that char R 2, we get, ( ) By Remark 1, this can be rewritten as, ( ) α α ( ) we get, ( ) for all (3) By the same method in Theorem 3, we complete the proof. Theorem 5: If for all, then. Replacing x by in, we get, ( ) for all (3) By using Theorem 3, we get. 137

5 Corollary 1: If for all, then. Theorem 6: If for all, then. Replacing x by in, we get, ( ) for all (3) By using Theorem 4, we get Corollary 2: that α is an endomorphism of R and f is a right α-centralizer such that α is not the identity map on U. If for all, then. The following example shows that the above results are not true in the case of arbitrary rings. Example: Suppose that S is any ring. Let and be a lie ideal of R. Define maps such that and Then, it is straightforward to see that f is a right α-centralizer of R, which is satisfy:(i) ([ ]) [ ], (ii), (iii) for all, however,. References: 1. Zalar, B On centralizers of semiprime rings. Comment.Math.Univ.Carolinae, 32(4), pp: Bresar,M On the distance of the composition of two derivations to the generalized derivations. Glasgow Math.J., 33, pp: Albas,E On τ-centralizers of semiprime rings. Siberian Math.J., 48, pp: Rehman,N On commutativity of rings with generalized derivations. Math. J. Okaya- ma Univ., 44, pp: Ashraf, M. and Ali, S On left multipliers and the commutativity of prime rings. Demonstratio Math., 41(4), pp: Ali,S. and Huang,S On left α- multipliers and commutativity of semiprime rings. Commun.Korean Math.Soc., 27, pp: Shuliang, H Generalized derivations of prime rings. International Journal of Mathematics and Mathematical Sciences, 2007, pp: Bergen,J.,Herstein,I.N. and Kerr,J.W Lie ideals and derivations of prime rings. Journal of Algebra, 71, pp: