Abstract

There has been considerable interest in the connection between the structure and the -structure of a ring, where denotes an involution on a ring. In this context, Oukhtite and Salhi (2006) introduced a new class or we can say an extension of prime rings in the form of -prime ring and proved several well-known theorems of prime rings for -prime rings. A continuous approach in the direction of -prime rings is still on. In this paper, we establish some results for -prime rings satisfying certain identities involving generalized derivations on -ideals. Finally, we give an example showing that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

1. Introduction

Throughout the paper, will denote an associative ring with center . For any , the symbol stands for the Lie product and the symbol denotes the Jordan product . A ring is called 2-torsion free, if whenever , with , then . Recall that a ring is prime if, for any implies   or . A ring equipped with an involution is to be -prime if   or . An example, according to Oukhtite and Salhi [1], shows that every prime ring can be injected in -prime ring and from this point of view -prime rings constitute a more general class of prime rings. An ideal is a -ideal if is invariant under ; that is, . Note that an ideal may not be a -ideal. Let be a ring of integers and . Consider a map defined by for all . For an ideal of , is not a -ideal of since .

Several authors have studied the relationship between the commutativity of a ring and the behavior of a special mapping on that ring. In particular, there has been considerable interest in centralizing automorphisms and derivations defined on rings (see, e.g., [2–4], where further references can be found). As defined in [5, 6], an additive mapping is called generalized derivation with associated derivation if Familiar examples of generalized derivations are derivations and generalized inner derivations and later included left multiplier, that is, an additive mapping satisfying for all . Since the sum of two generalized derivations is a generalized derivation, every map of the form , where is a fixed element of and , a derivation of , is a generalized derivation, and if has , all generalized derivations have this form.

In 2006, Oukhtite and Salhi [1] introduced a new class or we can say an extension of prime rings in the form of -prime ring. However, the actual motivation behind their first successful work came from Posner’s [7] second theorem only. In [8], they successfully extended the result for -prime ring. Recently, a major breakthrough has been achieved by Oukhtite and Salhi [9], where the important results by Posner, Herstein, and Bell have been proved for -prime rings. More precisely, Posner’s second theorem of existence of a nonzero centralizing derivation on prime ring which makes the ring commutative if is a prime ring of characteristic 2 with a nonzero derivation such that for all. Oukhtite and Salhi [9, Theorem 1.2] proved that the same result holds for -prime rings. Motivated by a well-known result by Herstein [10], Bell and Daif [11] studied derivation satisfying for all. This result has been extended for -prime rings [9, Theorem 1.3]. They initiated their work from [1], continued in [12, 13], and are spree of developing and extending more and more results which hold true for a prime ring. A continuous approach in this direction is still on. In this paper, we continue the study in the direction of Oukhtite by providing some results which are of independent interest and related to generalized derivations for -prime rings. More precisely, we will prove the following results.

Theorem 1. Suppose that is a 2-torsion free -prime ring and a -ideal of . If admits a generalized derivation with the additional condition that such that for all or for all , then is a left multiplier.

Theorem 2. Suppose that is a 2-torsion free -prime ring and a -ideal of . If admits a generalized derivation with the additional condition that such that for all , then .

Theorem 3. Suppose that is a 2-torsion free -prime ring and a -ideal of . If admits a generalized derivation with the additional condition that such that for all , then .

Theorem 4. Suppose that is a 2-torsion free -prime ring and a -ideal of . If admits a generalized derivation with the additional condition that such that or for all , then .

Theorem 5. Suppose that is a 2-torsion free -prime ring and a -ideal of . If admits a generalized derivation with the additional condition that such that or for all , then .

We will make extensive use of the following basic identities without any specific mention; for all ,(i), (ii), (iii).

2. Proof of Main Results

In all that follows, will denote the set of symmetric and skew-symmetric elements of ; that is, . We begin with the following results which will be used later to prove our theorems.

Lemma 6 (see [14, Lemma 3.1]). Let be a 2-torsion free -prime ring and let be a -ideal of . If such that , then   or .

Lemma 7 (see [9, Lemma 2.3]). Let be a 2-torsion free -prime ring and let be a -ideal of . If is a derivation on with the additional condition that such that , then .

Lemma 8 (see [14, Theorem 3.2]). Let be a 2-torsion free -prime ring and let be a -ideal of . If is a derivation on with the additional condition that such that , then is commutative.

Proof of Theorem 1. From hypothesis, we have Replacing with where in (2) and using it, we get Substituting for in (3) and using (3), we arrive at Putting in (4), we obtain This implies that For all , as commutes with then (6) is forced to and Lemma 6 yields or . Let ; since , then we have or . If , then (6) implies Equations (7) and (6), by Lemma 6, imply or . Similarly, if , then and using (6) we have for all . Once again using Lemma 6, the last equality together with (6) leads to or . Hence, in both cases we have or for all . Since a group cannot be the union of proper subgroups, according to Brauer’s trick, either or . Taking together with Lemma 7, we are forced to consider that Then, we conclude, by Lemma 8, that is commutative. Hence, our hypothesis becomes Using 2-torsion freeness, we have so that Since is prime and , (11) yields for all and so by simple calculation. From , it then follows that for all , , and therefore by Lemma 6, which is a contradiction. Consequently, we take and so by Lemma 7. Hence, this completes the proof.

Similar arguments can be adapted in the case and we can omit the similar proof.

Proof of Theorem 2. From hypothesis, we have Taking instead of in (12) and using it, we obtain Replacing with in (13) and in view of (13), we get Using 2-torsion freeness, we have so that For all , as commutes with , then (16) forces and from Lemma 6, the last equality yields for all or . Since a group cannot be the union of proper subgroups, according to Brauer’s trick either for all or . Supposing that and by using similar reasoning to the one in [1, Proof of Theorem 1.1] yield that is commutative. Hence, using 2-torsion freeness, our hypothesis becomes ; that is, for all . From , it follows that for all . Therefore, It follows from Lemma 6 that , which is a contradiction. Consequently, we take and acts as a left multiplier. In this situation, our hypothesis is forced to Clearly, and also we obtain , comparing these two expressions for gives us for all by 2-torsion freeness. That is, for all which leads to by Lemma 6. This completes the proof.

Proof of Theorem 3. From hypothesis, we have Taking instead of in (19) and using it, we obtain Replacing with in (20) and using it, we have Since is 2-torsion free, we find so that The last equality is the same as (16) in the proof of Theorem 2. Thus, by using the same arguments as in the proof of Theorem 2, we can conclude the result here.

Proof of Theorem 4. From hypothesis, we have Replacing by in (24) and using it, we find Taking instead of in (25) and using it, we get so that Equation (27) is the same as (16) in the proof of Theorem 2. Thus, by using the same arguments as in the proof of Theorem 2, we can conclude the result here.
Similar arguments can be adapted in the case and we can omit the same proof.

Proof of Theorem 5. From hypothesis, we have Replacing with in (28) and using it, we find Putting in (29) and using it, we get so that The last equality is the same as (27) same as (16) in the proof of Theorem 4. Thus, by using the same arguments as in the proof of Theorem 4, we can conclude the result here.
Application of similar arguments to the one mentioned above yields the case and we can omit the similar proof.

3. Homomorphism or Antihomomorphism

Suppose are two endomorphisms of . An additive mapping is called generalized -derivation associated with -derivation , where is defined as for all if for all . Obviously, every -derivation on is just a generalized derivation on , where is the identity mapping. If is a generalized -derivation of and   or for all , then is called a generalized -derivation which acts as a homomorphism or antihomomorphism on , where is a nonempty subset of .

Recently, Bell and Kappe [15] proved that if is a derivation of prime ring which acts as a homomorphism or an antihomomorphism on a nonzero ideal of , then on . Thereafter, Albaş and Argaç [2] extended this result to generalized derivation. Further, Oukhtite and Salhi [1] proved that the above result is also true for -prime rings. It is natural to raise a question: is the above result valid in generalized -derivation? In this context, we give an affirmative answer to the question.

Theorem 9. Suppose that is a 2-torsion free -prime ring, is a -ideal, and is a generalized -derivation with the additional condition that , where is an automorphism of such that . If acts as a homomorphism or as an antihomomorphism on , then   or .

Proof
Step I. Let act as a homomorphism on ; then we have Replacing with in (32), we get so that Since is a homomorphism on , we have From (34) and (35), we have and hence . Letting , it is easy to see that is a nonzero -ideal; that is, Now (37) yields . As commutes with and is a -ideal, then, by Lemma 6, we have either   or   for all ; namely,   or   on .
Step II. Letting act as an antihomomorphism on , then we have Replacing by in (38), we find so that Putting in (40) and using it, we arrive at As is an automorphism, that is, if we set , (41) yields For all and from (42), we have In view of Lemma 6, the last equality yields   or  . Now, assume that . Then is a left multiplier and (38) yields Taking instead of in (44) and using it, we reach the following: As is a -ideal and from Lemma 6, we get Comparing (44) and (46), we have Therefore, in both cases we find for all . Thus, acts as a homomorphism and hence, by the previous part of this theorem, we conclude that .

4. Counter Example

Here, we try to construct some examples to demonstrate that the above results are not true in the case of arbitrary ring.

Example 1. Let where is a ring of integer and We define the following maps: Then it can be seen easily that is a -ideal of with involution and is a generalized derivation with additional condition that . Moreover, it is straightforward to check that satisfies the following properties: (i) , (ii) , (iii) , (iv) , (v) for all . However, neither nor is a left multiplier.