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International Journal of Mathematics and Mathematical Sciences

Volume 2014, Article ID 216039, 8 pages

http://dx.doi.org/10.1155/2014/216039
Research Article

Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

1Department of Mathematics and Computer Science, University of Messina, Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina, Italy

2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

3Faculty of Science, Islamic University in Madinah, P.O. Box 170, Madinah 42351, Saudi Arabia

Received 3 January 2014; Accepted 12 May 2014; Published 3 June 2014

Academic Editor: Kaiming Zhao

Copyright © 2014 Vincenzo De Filippis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a 2-torsion free ring and let be a noncentral Lie ideal of , and let and be two generalized derivations of . We will analyse the structure of in the following cases: (a) is prime and for all and fixed positive integers ; (b) is prime and for all and fixed integers ; (c) is semiprime and for all and fixed integer ; and (d) is semiprime and for all and fixed integer .

1. Introduction

Let be an associative ring with characteristic different from 2, its center, its (right) Utumi quotient ring, and its extended centroid. The simple commutator will be denoted by . Recall that a derivation is an additive map satisfying the product rule for all .

A left multiplier on a ring is an additive map satisfying the rule for all . In case there exists an endomorphism of such that for all , then is called left -multiplier of .

A generalized derivation on a ring is an additive map satisfying for all and some derivation of . A significative example is a map of the form , for some ; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from the algebraic point of view might be interesting (see, e.g., [1]). Notice that any derivation is a generalized one and also that the generalized inner derivations include left multipliers and right multipliers. Thus the concept of generalized derivation covers both the concept of derivation and the concept of left (right) multipliers.

Since the sum of two generalized derivations is a generalized derivation, of course every map of the form is a generalized derivation on , where is a fixed element of and is a derivation of .

In [1, Theorem 3] Lee proved that every generalized derivation on a dense right ideal of can be uniquely extended to the Utumi quotient ring of , and thus any generalized derivation of can be defined on the whole ; moreover it is of the form for some and is a derivation on ( is said to be a generalized derivation associated with derivation ).

Many results in the literature indicate that the global structure of a ring is often tightly connected to the behaviour of additive mappings defined on .

In [2] Bergen proved that if is an automorphism of such that , for all , where is a fixed integer, then . Daif and Bell [3] showed some results which have the same flavour, when the automorphism is replaced by a nonzero derivation . In [3] it is proved that if is a semiprime ring with a nonzero ideal such that , or , for all , then is central. Later Hongan [4] proved that if is a 2-torsion free semiprime ring and a nonzero ideal of , then is central if and only if , or , for all . Recently in [5] Ashraf and Ali obtained commutativity theorems for prime rings admitting left multipliers which satisfy similar conditions. More precisely in [5] it is showed that a prime ring must be commutative if there exist a nonzero ideal of and a left multiplier , which is not the identity map on , such that one of the following holds: (i) for all [5, Theorem ]; (ii) for all [5, Theorem 3.1]; and (iii) for all [5, Theorem 3.3]. Moreover the same results hold in case is replaced by a generalized derivation of .

In a more recent paper [6], Ali and Huang extended the previous cited results in the case is a semiprime ring, is a nonzero ideal of , and is a left -multiplier of . They proved that if one of the following holds: (i) for all [6, Theorem 2.1]; (ii) for all [6, Theorem 2.2]; (iii) for all [6, Theorem 2.3]; and (iv) , for all [6, Theorem 2.5]. Moreover, in case is a prime ring, the same conditions force the commutativity of .

A natural question is to consider additive maps such that , when and are both either monomials or powers of the commutator . In this sense, in [7] it is proved that, under appropriate torsion assumptions, a prime ring is commutative if it admits a nonzero derivation satisfying one of the following: (i) ; (ii) ; (iii) ; and (iv) , for all .

So it seems natural to ask about the case when the derivation is replaced by a generalized derivation. Motivated by the previous cited results, in this paper we will introduce two different generalized derivations acting on and satisfying some appropriate conditions on some suitable subsets of . We will prove the following.

Theorem 1. Let be a 2-torsion free prime ring and let and be two generalized derivations associated with derivations and , respectively. Suppose that there exist integers such that for all . Then either satisfies the standard identity or one of the following holds:(a) is an inner ordinary derivation of and for all there exists such that ;(b) and , for all .

Theorem 2. Let be a 2-torsion free prime ring and let be a noncentral Lie ideal of , and let and be two nonzero generalized derivations associated with derivations and , respectively. Suppose that there exist integers such that for all . Then satisfies the standard identity .

In the last section we study some commutativity conditions for a semiprime ring with a generalized derivation satisfying suitable algebraic conditions. More precisely, we will prove the following.

Theorem 3. Let be a 2-torsion free semiprime ring and let and be generalized derivations associated, respectively, with derivations and . Set and such that for all . Then there exists a central idempotent of   such that, on the direct sum decomposition , the generalized derivations and vanish identically on and the ring satisfies .

Theorem 4. Let be a 2-torsion free semiprime ring and let and be generalized derivations associated with derivations and , respectively. Set such that for all . Then contains a nonzero central ideal, , , and there exist such that , , for all .

Moreover there exists a central idempotent of   such that, on the direct sum decomposition , the generalized derivations and vanish identically on and the ring is commutative.

2. Action of Generalized Derivations on Prime Rings

In order to prove the main result in this section, we first fix some intermediate lemmas and theorems. We begin with the following.

Remark 5 (see [8], Lemma 3). Let be a prime ring with extended centroid and let be any polynomial, not necessarily multilinear. Then the right (left) annihilator of is zero, unless when is an identity for .

Lemma 6. Let be a primitive ring which is isomorphic to a dense ring of linear transformations of a vector space over the field such that and let , , different integers. If for any , , then and for all , unless when .

Proof. Let . Since , there exists such that are linearly -independent. By the density of there exist such that so that and by the hypothesis that is , which implies . Hence for all and by Remark 5 we have that either or for any .

Lemma 7. Let be a 2-torsion free prime ring, a nonzero two-sided ideal of , and a noncentral polynomial over , the extended centroid of . If is a generalized derivation associated with a derivation such that for all , then , for all .

Proof. In light of previous remarks, we have that there exists , the Utumi quotient ring of , such that , for all . Thus satisfies the generalized differential identity

Since, by [9], and satisfy the same differential identities, then we have that satisfies . Let be the additive subgroup generated by the subset

is a Lie ideal of ; indeed for any , one has

If is noncommutative then, by [10, pages 4-5], there exists a nonzero two-sided ideal of such that . In this case it is easy to see that , for all . In particular for all . Since is a noncentral Lie ideal of , it follows easily that must be zero (see, e.g., Theorem 3.3 in [11]).

Hence we may consider it the only case when is commutative.

Thus is an identity in . This means that there exist a field and a positive integer such that is also an identity in . If , is commutative, thus we suppose .

Since is not central valued on , there exist such that , so that is also a generalized identity in . By a result of Lee (see [12], Theorem), we have the contradiction that is central valued on .

Lemma 8. Let be a 2-torsion free primitive ring which is isomorphic to a dense ring of linear transformations of a vector space over the field such that , and let , , , different integers. If for any , , then , and one of the following holds:(a) ;(b)for all there exists such that .

Proof. Let such that are linearly -independent. Since , there exists such that are linearly -independent. By the density of there exist such that so that , and by the hypothesis which implies . Thus for all and by Lemma 7 we get the required conclusions.

Assume now that, for any , are linearly -dependent. In this case standard arguments show that ; hence and by Lemma 6 we get ; that is, and .

Lemma 9. Let be a 2-torsion free primitive ring which is isomorphic to a dense ring of linear transformations of a vector space over the field such that , and let and be such that , for all . Then one of the following holds:(a) , and ;(b) , and for all there exists such that ;(c) and for all .

Proof. Suppose first that is finite. In this case , the ring of all matrices over with . Let be the usual matrix unit and denote , , with . Fix and choose . By the hypothesis and left multiplying by , for any , it follows that . This means that the matrix is diagonal. Let now be any automorpism of and note that , for all . Therefore must be a diagonal matrix. In particular, for any , must be a diagonal matrix. By easy computation it follows that ; that is, . Analogously one can prove that .

Therefore we have that

Suppose now that . By Lemma 2 in [13], satisfies the following generalized identity . Let be any idempotent minimal element and recall that is generated by such idempotent elements. Since satisfies , then in particular . Thus we can easily obtain both and , which imply and ; that is, and . Also in this case there exist such that and . Therefore in any case Identity (9) holds and the conclusion follows as an application of Lemma 8.

Lemma 10. Let be a 2-torsion free prime ring, , and let be such that , for all . Then either satisfies , the standard identity of degree , or one of the following holds:(a) , and ;(b) , and for all there exists such that ;(c) and for all .

Proof. Firstly assume that does not satisfy any nontrivial generalized polynomial identity. In light of [14] and by our assumption, it follows that both are trivial generalized polynomial identities of . This means that , so that is a trivial generalized polynomial identity for .

Hence and .

Consider now the case that satisfies some nontrivial generalized polynomial identity. By Theorem 3 in [15] it follows that is a primitive ring with , where is the extended centroid of , and the Utumi quotient ring is a -algebra centrally closed. Since and satisfy the same generalized polynomial identities (see [14]), without loss of generality, we may replace by and by and is a -algebra centrally closed. Then is a dense ring of linear transformations of a vector space over . In case , then satisfies the standard identity . In case we may apply Lemma 9. In any case we are done.

We consider now the more general situation.

Proof of Theorem 1. In all that follows we assume that does not satisfy ; if not we are done.

Since any generalized derivation of can be implicitly assumed to be defined on the whole and assumes the form for some and a derivation on , we may assume that there exist and derivations on such that

Since and satisfy the same generalized polynomial identities [14] as well as the same differential identities [9], then, without loss of generality, to prove our results we may assume that where , are derivations on . We divide the proof into cases.

Case 1. Let and be both inner derivations in , so that and , for suitable elements . Thus we have , for all . In this case we apply Lemma 10 and we have that either satisfies or one of the following holds:(a) and ; that is, ;(b) , , and ; that is, is the inner derivation induced by the element ; moreover for all there exists such that ;(c) , for all (i.e., ); moreover for a suitable . In this case, our assumption implies that for all , which means , since is not commutative. Therefore and .

Case 2. Assume that and are -independent modulo -inner derivations.

By the main assumption, satisfies

Then by the result in [16], we have that satisfies

In particular satisfies the blended component

Thus is a PI-ring and there exists satisfying (15). For , , and we have the contradiction .

Case 3. Assume finally that and are not both inner derivations (otherwise we are done by Case 1) such that and are -dependent modulo -inner derivations. Without loss of generality say ; that is, , for suitable and . Under this assumption, we prove that a number of contradictions follow. We may assume the following:(i) are both nonzero derivations of . In fact, if either or is zero then both and are inner generalized derivations of , which is a contradiction;(ii) is not an inner derivation, if not both and are inner generalized derivations of , which is again a contradiction.

Suppose first that ; then . Thus satisfies that is,

By [16] it follows that satisfies and in particular satisfies the blended component

Therefore is a PI-ring, so that there exists a field such that and the matrix ring satisfy the same polynomial identities. Moreover, we may assume , since does not satisfy . Notice that if we choose , , and then the contradiction follows.

Then assume . Thus satisfies that is,

By [16] it follows that satisfies and in particular satisfies both

As above, is a PI-ring, so that there exists a field such that and the matrix ring satisfy (24). Since we may assume that , then for , , , and in (24) we get ; that is, . Moreover, by (23) and using Lemma 10 we have in any case , that is, , and one of the following holds. (a) for all and there exists such that . In this case we have and for all . By the main assumption we also get , for all . Hence , which implies and .(b) and . This implies , for all . Since , by Lemma 7 we have that for all there exists such that . Without loss of generality we consider and let be any nonzero element of . Denote and . Thus there exist such that and . Hence implying

In other words, for all and for all , we have . In case then for all and as above there exists a field such that and the matrix ring satisfy . Since we may assume that , a contradiction follows easily. Thus for any . In this last case and by (25), it follows that , for any . Hence, for , we get ; that is, for any choice of , which means for all .

Corollary 11. Let be a 2-torsion free prime ring and let be a noncentral Lie ideal of , and let and be two generalized derivations associated with derivations and , respectively. Suppose that there exist integers such that for all . Then either satisfies the standard identity or one of the following holds:(a) is an inner ordinary derivation of and for all there exists such that ;(b) and , for all .

Proof. Since is not central in , then by [10, pages 4-5] there exists a nonzero two-sided ideal of such that . Therefore for all . As above we write for suitable and derivations of . Thus satisfies the differential identity

Since , , and satisfy the same differential identities (see [9]), then (27) is satisfied by . Hence we conclude by Theorem 1.

Proof of Theorem 2. Firstly we notice that for we have , for all . By Corollary 11 we have that is a PI-ring and, if assumed that does not satisfy , then one has . As remarked above, there exists an ideal of such that ; therefore for all . Since and satisfy the same differential identities, we also have that (28) is satisfied by . By Lemma 7, and since , is central for all , moreover there exists a field such that and satisfy the same polynomial identities. Of course we assume that , since does not satisfy . Fix and .

Thus where . Hence which is a contradiction.

Corollary 12. Let be a 2-torsion free prime ring, its Utumi quotient ring, its extended centroid, noncentral Lie ideal of , and and two generalized derivations associated with derivations and , respectively. Suppose that there exists integer such that for all . Then either or satisfies the standard identity and is an ordinary derivation of .

Proof. For in our main assumption we get ; moreover by applying Theorem 2 it follows that must satisfy the standard identity .

Therefore and we may assume that , the matrix ring over . Since is not central and is 2-torsion free, as remarked above, it follows that . Hence we have that is satisfied by . Let such that and , for all ; then satisfies the generalized differential identity

Firstly notice that in case then satisfies and by Lemma 7 it follows that unless is central valued on . In the latter case for and a contradiction follows.

Thus we assume both and and denote (of course is a derivation of ). Here we apply again Kharchenko’s theory, using the fact that is central valued on , and we prove that . To do this, we divide the proof into two cases.

Case  1. Let be the inner derivations in . Thus we have that , for all , which implies .

Case  2. Assume that is not inner. By Kharchenko’s theory in [16], and since satisfies then satisfies in particular is a generalized identity for , which implies again that .

Corollary 13. Let be a 2-torsion free prime ring, its Utumi quotient ring, its extended centroid, and and two generalized derivations associated with derivations and , respectively. Suppose that there exists integer such that for all . Then either or is commutative.

Proof. By Corollary 12, if we assume that and , it follows that and is ordinary derivation of . In particular for all and by Lemma 7 it follows that either or , for all .

In case , then and for all . Again by Lemma 7, since , one has ; in particular for and , it follows the contradiction . Therefore must be commutative.

Let , so that , for all follows by the previous argument. Thus is commutative and by the main assumption it follows . Once again by Lemma 7, since , it follows , which leads to a contradiction, as remarked above.

An easy consequence of Corollary 13 is the following.

Corollary 14. Let be a 2-torsion free prime ring, its Utumi quotient ring, its extended centroid, and a generalized derivation of . Suppose that there exists integer such that for all . Then either or is commutative.

3. Results in Semiprime Rings

In order to prove the main result of this section we will make use of the following facts.

Remark 15. Let be a semiprime ring and let be a generalized derivation of associated with derivation . If , then .

Claim 1 (see [17, Proposition 2.5.1]). Any derivation of a semiprime ring can be uniquely extended to a derivation of its left Utumi quotient ring , and so any derivation of can be defined on the whole .

Claim 2 (see [18, page 38]). If is semiprime then so is its left Utumi quotient ring.

The extended centroid of a semiprime ring coincides with the center of its left Utumi quotient ring.

Claim 3 (see [18, p. 42]). Let be the set of all the idempotents in , the extended centroid of . Assume that is B-algebra orthogonal complete. For any maximal ideal of , forms a maximal prime ideal of , which is invariant under any derivation of .

We are now ready to prove Theorems 3 and 4.

Proof of Theorem 3. Let and derivations of be such that and , for all . By Claim 2, , and by Claim 1   and can be uniquely defined on the whole . Since and satisfy the same generalized differential identities, then , for all . Let be the complete Boolean algebra of idempotents in and any maximal ideal of .

Since is -algebra orthogonal complete (see [18, page 42, of Fact 1]), by Claim 3, is a prime ideal of , which is both -invariant and -invariant. Let and be the derivations induced, respectively, by and on and denote , . For any , . In particular is a prime ring and so, by Corollary 12, either and in or satisfies . This implies that, for any maximal ideal of , and or , for all . In any case and also . From [17, Chapter 3] there exists a central idempotent element of , such that , and satisfies .

Proof of Theorem 4. Here we repeat the same argument above. Let such that and . Let be the complete Boolean algebra of idempotents in and any maximal ideal of .

Since is -algebra orthogonal complete (see [18, p. 42, (2) of Fact 1]), by Claim 3, is a prime ideal of , which is both -invariant and -invariant. Let and be two derivations induced, respectively, by and on and denote , . For any , . Since is a prime ring so, by Corollary 13, either and in or is commutative. Moreover, if and in then, by Remark 15, it follows that and in .

This implies that, for any maximal ideal of , one of the following holds:(a)either , , and ;(b)or .

In any case and imply that . Analogously and implying that .

As mentioned above it follows that there exists a central idempotent element in such that, on the direct sum decomposition , and vanish identically on and the ring is commutative.

Moreover, implies ; that is, both and . Therefore and, by the semiprimeness of , it follows ; that is, . Analogously one can prove that .

On the other hand, since by Theorem 3 in [9] and satisfy the same differential identities, then and , which imply that , , and contains some nonzero central ideals, unless when and .

In the last case, for all , and in particular for all . The semiprimeness of forces (since ), so that for all . Finally by Theorem 1 in [19], the commutativity of follows which is a contradiction.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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