Abstract

We extend some well known commutativity results concerning a nonzero square closed -Lie ideal and generalized -derivations of -prime rings.

1. Introduction

Let be an associative ring with center . For any the symbol represents commutator . Recall that a ring is prime if implies or . An additive mapping is called an involution if and for all . A ring equipped with an involution is called a ring with involution or -ring. A ring with an involution is said to be -prime if or implies that or . Every prime ring with an involution is -prime but the converse need not hold, in general. An example introduced by Oukhtite and Salhi [1] justifies the above statement; that is, is a prime ring and where is the opposite ring of . Define involution on as . is -prime, but not prime. This example shows that -prime rings constitute a more general class of prime rings. All that follows the symbol , first introduced by Oukhtite and Salhi, will denote the set of symmetric and skew symmetric elements of , that is, .

An additive subgroup of is said to be a Lie ideal of if . A Lie ideal is said to be a -Lie ideal if . If is a Lie ( -Lie) ideal of , then is called a square closed Lie (-Lie) ideal of if for all . An additive mapping is called a derivation if holds for all . For a fixed , the mapping given by is a derivation which is said to be an inner derivation. An additive mapping is called a generalized derivation if there exists a derivation such that This definition was given by Bresar in [2].

Let be the ring of integers. Set and . We define the following maps on , , and . Then it is easy to see that is a nonzero square closed -Lie ideal of and is a generalized derivation associated with a nonzero derivation commuting with .

Let and be any two automorphisms of . An additive mapping is called an -derivation if holds for all . Inspired by the definition of an -derivation, the notion of a generalized derivation was extended as follows. Let and be any two automorphisms of . An additive mapping is called a generalized -derivation on if there exists an -derivation such that Of course a generalized -derivation is a generalized derivation on , where is the identity mapping on .

Several authors studied commutativity in prime and semiprime rings admitting derivations and generalized derivations which satisfy appropriate algebraic conditions on suitable subsets of the rings. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from algebraic point of view might be interesting. Recently, some well-known results concerning prime rings have been proved for -prime ring by Oukhtite and Salhi (see [1], where further references can be found).

In [3], Daif and Bell showed that the ideal of a semiprime ring is contained in the center of if any of the following conditions is satisfied: Several authors investigated this result for prime or semiprime ring admitting derivation or generalized derivation (see [4–6]). Also, Ashraf et al. proved some commutativity theorems for prime rings in [7, 8]. Motivated by these theorems, we will investigate some commutativity theorems for a nonzero square closed -Lie ideal and generalized -derivations of -prime rings.

2. Preliminaries

Throughout the paper, will be a -prime ring with characteristic not two and which commute with . will be a nonzero generalized -derivation of with associated nonzero -derivation which commutes with and will be a nonzero -Lie ideal of . Also, we will make extensive use of the basic commutator identities:

Lemma 1 (see [1, Lemma 4]). Let be a -prime ring with characteristic not two and let be a nonzero -Lie ideal of and , . If or then or or .

Lemma 2 (see [9, Lemma 2.7]). Let be a -prime ring with characteristic not two and let be a nonzero -Lie ideal of . If such that , then either or .

The following lemma is an immediate consequence of Lemma 2.

Lemma 3. Let be a -prime ring with characteristic not two and let be a nonzero -Lie ideal of . If , then .

Lemma 4 (see [10, Lemma 9]). Let be a -prime ring with characteristic not two and let be a nonzero -derivation of which commutes with and let be a nonzero -Lie ideal of . If , commutes with and , then .

Lemma 5 (see [10, Theorem 2]). Let be a -prime ring with characteristic not two and let be a nonzero -derivation of which commutes with and let be a nonzero -Lie ideal of . If , commutes with and , then .

3. Results

Theorem 6. If satisfies any one of the following conditions,(i), for all ,(ii), for all ,then .

Proof. (i) If , then for all , . Replacing by , , we getUsing and the above equation in the last equation, we getthat is,Hence we haveSince is a nonzero -Lie ideal of , the above yieldsTherefore, we conclude thatBy Lemma 1, we get either , for all or , for all . And so, , for all . By Lemma 2, we have , for all or . Thus, we get .
We shall assume that . We are given thatReplacing by in (11) and using , we haveThis implies thatBy the hypothesis, we obtainSubstituting for in (14) and using , we getthat is,Since is a nonzero -Lie ideal of , we obtainthat is,By Lemma 1, we arrive at either , for all or for each . Let , as , , and , for all or . Hence, we have or , for all . We setClearly each of and is additive subgroup of . Moreover, is the set-theoretic union of and . But a group cannot be the set-theoretic union of two proper subgroups; hence or . In the former case, . By Lemma 3, we obtain . In the latter case, we have by Lemma 4. Thus the proof is completed.
(ii) If satisfies , for all , then satisfies the condition and hence by part (i), our result follows.

Theorem 7. If satisfies any of the following conditions,(i), for all ,(ii), for all ,then .

Proof. (i) If , then , for all . Replacing by , in the above equation and using , we obtainNow repeating the similar arguments as we used in the proof of Theorem 6, we get the required result.
Henceforth, we assume that . For any , we haveReplacing by in (21) and using , we havethat is,By application of (21), we find thatTaking for in (24) and using , (24), we getthat is,The last expression is the same as (16) and hence the results follow.
(ii) Using similar techniques with necessary variations, we get the required result.

Theorem 8. If  , for all , then .

Proof. We haveReplacing by in (27) and using , we getBy the hypothesis, we obtainSince and is an automorphism of , we haveHence we arrive atSince is a nonzero -Lie ideal of , the above yieldsTherefore, we getBy Lemma 1, we get either , for all or , for all . Hence we conclude thatSubstituting for in (34) and using , (34), we getReplacing by in the last equation, we getTaking instead of in the above equation and using this, we havethat is,Using Lemma 1 and the similar arguments as we used after (16), we get the required result from (18).

Theorem 9. If  , for all , then .

Proof. Substituting , for in the hypothesis and using , we obtainAgain using the hypothesis we haveReplacing by , in (40) and applying this equation, we arrive atsoSince is a nonzero square closed -Lie ideal of , we have is a nonzero square closed -Lie ideal of , too. Hence using , we getBy the application of Lemma 1, we get either , for all or , for each . Since is an automorphism of , we have either , for all or , for each . Let , as , and , for all or . Hence, we have or , for all , .
Now, we setClearly each of and is additive subgroup of . Moreover, is the set-theoretic union of and . But a group cannot be the set-theoretic union of two proper subgroups; hence or . In the former case, we have for all . Using Lemma 2, we get for all or . By Lemmas 4 and 3, we have . In the latter case, ; that is, by Lemma 5, so again using Lemma 3, we get . This completes the proof.

Theorem 10. If , for all , then .

Proof. If , then , for all . Replacing by in this equation and using , we see thatSince is a nonzero -Lie ideal of , we getBy Lemma 1, we have or , for all , , and so .
Now, we assume . By the hyphothesis, we getReplacing by , in (47) and using our hypothesis, we getAgain replacing by , in the last equation and using , we obtainTaking instead of in (49) and using this equation, we havesoBy further application of similar arguments as used in the end of proof of Theorem 9, we get the required result.

Theorem 11. If satisfies any of the following conditions,(i), for all ,(ii), for all ,then .

Proof. If , then , for all , . Now using the similar arguments as used in the proof of Theorem 7, we get the required result.
Suppose that andReplacing by , and using the fact that , we haveReplacing , for in (53) we getUsing (49) in the last equation, we getsoSince is automorphism of , we obtainUsing , and is a nonzero -Lie ideal of , we haveBy Lemma 1, we get either or , for all , , . If , for all , then we have by Lemma 2. Hence we conclude that or , so in both cases; thus we get by Lemma 4.

Theorem 12. If , for all , then .

Proof. If , then , for all . In particular, , for all and henceReplacing by , we get , for all , , . Since , we see thatthat is,Since is a nonzero -Lie ideal of , we get , for all , . By Lemma 1, we get or for all , . This implies that , for all . Let , as , , and . Hence we have , for all . By Lemma 2, we find that . Thus, we get .
Hence, onward we assume that . By the hypothesis, we haveThis can be rewritten asReplacing by in (63) and using , we arrive atCommuting this term with and using the hypothesis, we obtainsoReplacing , by in this equation and using this, we getsoBy trivial modification of the argument in the proof of Theorem 8, we get the required result.

Theorem 13. If for all , then .

Proof. If is a generalized -derivation satisfying the property for all , then satisfies the condition , for all . Hence we have by Theorem 12.

Conflict of Interests

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