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Volume 2012 (2012), Article ID 729356, 11 pages
Commutativity Theorems for *-Prime Rings with Differential Identities on Jordan Ideals
1Département de Mathématiques, Faculté des Sciences et Techniques, Université Moulay Ismaïl, BP 509, Boutalamine, Errachidia 52000, Morocco
2Department of Mathematics, College of Sciences, Taibah University, P.O. Box 2285, Madina, Saudi Arabia
Received 20 November 2012; Accepted 12 December 2012
Academic Editors: P. Damianou, M. Fontana, and M. Siles Molina
Copyright © 2012 A. Mamouni 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.
In this paper we explore commutativity of -prime rings in which derivations satisfy certain differential identities on Jordan ideals. Furthermore, examples are given to demonstrate that our results cannot be extended to semiprime rings.
Throughout this paper, will represent an associative ring with center . is -torsion free if yields . We recall that is prime if implies or . A ring with involution is -prime if yields or . It is easy to check that a -prime ring is semiprime. Moreover, every prime ring having an involution is -prime but the converse does not hold, in general. For example, if denotes the opposite ring of a prime ring , then equipped with the exchange involution , defined by , is -prime but not prime. This example shows that every prime ring can be injected in a -prime ring and from this point of view -prime rings constitute a more general class of prime rings.
In all that follows will denote the set of symmetric or skew-symmetric elements of . For and . An additive subgroup of is a Jordan ideal if for all and . Moreover, if , then is called a -Jordan ideal. We will use without explicit mention the fact that if is a Jordan ideal of , then and [1, Lemma 1]. Moreover, From  we have , and for all .
A mapping is called strong commutativity preserving on a subset of if for all . An additive mapping is called a derivation if holds for all pairs . Recently, many authors have obtained commutativity theorems for -prime (prime) rings admitting derivation, generalized derivation, and left multiplier (see [3–8]). In this paper, we will explore the commutativity of -prime rings equipped with derivations satisfying certain differential identities on Jordan ideals.
2. Differential Identities with Commutator
We will make some use of the following well-known results.
Remarks 2.1. Let be a -torsion free -prime ring and a nonzero -Jordan ideal.(1) (see [6, Lemma 2]) If , then or .(2) (see [6, Lemma 3]) If , then .(3) (see [7, Lemma 3]) If , then is commutative.(4) (see , Lemma 3]) If is a derivation such that for all , then .
We leave the proofs of the following two easy facts to the reader.(5) If , then . In particular, if or , then .(6) If admits a derivation such that , then .
Lemma 2.2. Let be a 2-torsion free -prime ring and a nonzero -Jordan ideal. If admits a nonzero derivation such that for all , then is commutative.
Proof. First suppose that . From it follows that
Substituting for in (2.1), where , we obtain which leads to
For , (2.2) together with Remarks 2.1(1) forces , in which case , or . Since , in both the cases we arrive at
Let ; using the fact that and , we obtain . Replacing by in (2.2), we get which combined with (2.2) yields either or so that . This implies that
and hence by Remarks 2.1(4), which contradicts our hypothesis and thus .
Now let ; from it follows that for all and thus either or . If , then using similar arguments as used in [5, Proof of Theorem 3] we conclude that is commutative. Assume that ; the fact that yields . Similarly in view of , we find that and therefore Replacing by in (2.5) and using (2.5) we get . Again, replace by in the last equation, to get so that In view of -primeness, (2.6) assures that either or .
If for all , then by Remarks 2.1(4) contradiction, thus for all and , then [5, Proof of Theorem 3] implies that is commutative.
Theorem 2.3. Let be a -torsion free -prime ring and a nonzero -Jordan ideal. If admits a nonzero derivation , commuting with , such that either is strong commutativity preserving on or for all , then is commutative.
Proof. (i) Assume that is strong commutativity preserving on . In this case the condition is not necessary. Indeed, if , then and Remarks 2.1(2) forces . Thus is commutative by Remarks 2.1(3).
We are given that Replacing by in the above expression, where , we get Again, replacing by in (2.8), where and , and using (2.8) we obtain Putting for in (2.9), where we find that for all and . Substituting for in (2.10), then we have and therefore Let ; from (2.11) it follows that , so either or .
Suppose that Replacing by in (2.12), where , we find that and therefore Since , then (2.13) assures that In view of (2.13) and (2.14), the -primeness of forces for all or for all .
Suppose that thus Replacing by in (2.15) we arrive at and thus which leads to or . Hence, in both cases we find that for all and [5, Proof of Theorem 3] assures that is commutative, thereby for all .
In conclusion, Let ; as , from the above relation it follows that or for all .
Assume that . Hence (2.11) can be rewritten as Combining the latter relation with (2.11), we get or which makes it possible to conclude, using similar arguments as above, that or .
Now suppose . In relation to (2.11) let be ; then we have which gives, because commutes with , Whence it follows, according to (2.11), that or . Thus in both cases we find that Now let and . Clearly, and are additive subgroups of whose union, because of (2.20), is Hence, by Brauer's trick, either or . If , then and hence, by Remarks 2.1(6), thus , so for all , whence it follows, according to Lemma 2.2, that is commutative.
(ii) Assume that Substituting for in (2.21), where , we get Since (2.22) is the same as (2.8), reasoning as in the first case, we conclude that is commutative.
In  Herstein proved that if is a prime ring of characteristic not 2 equipped with a nonzero derivation such that for all , then is commutative. As an application of the above theorem, we get the following theorem which generalizes Herstein's result for Jordan ideals.
Theorem 2.4. Let be a -torsion free prime ring and a nonzero Jordan ideal. If admits a nonzero derivation such that for all , then is commutative.
Proof. Let be the additive mapping defined on by . Clearly, is a nonzero derivation of . Moreover, if we set , then is a -Jordan ideal of such that for all . Since commutes with and is -prime, then Theorem 2.3 assures that is commutative and thus so is .
An application of similar arguments yields the following.
Theorem 2.5. Let be a 2-torsion free prime ring and a nonzero Jordan ideal. If admits a derivation strong commutativity preserving on , then is commutative.
In 2010, Oukhtite et al. [8, Theorem 2] established that if a -torsion free -prime ring admits a nonzero derivation such that for all in a nonzero square closed Lie ideal , then or . Motivated by this result, our aim in the following theorem is to explore the commutativity of -prime rings admitting a nonzero derivation satisfying the above condition on -Jordan ideals.
Theorem 2.6. Let be a 2-torsion free -prime ring and a nonzero -Jordan ideal. If admits a nonzero derivation which commutes with such that for all , then is commutative.
Proof. Suppose that
Replacing by in (2.23) where and , in light of , we find that
Again replacing by in (2.24) with and , we get
Substituting by in (2.25) with and employing (2.25), we obtain
Replacing by in (2.26), where , then we have
for all and for all . Putting in (2.27), we find that
Let ; from (2.29) it follows, in light of -primeness, that
Replacing by in the above expression, we get and thus
being -prime implies that either or .
If , then As (2.33) is the same as (2.15), then reasoning as in the proof of Theorem 2.3 we find that is commutative and therefore .
In conclusion, Let ; as , in view of (2.34) we obtain that either or .
If , then and hence, because commutes with , (2.29) reduces to In light of (2.29), the latter expression together with -primeness of shows that either or which, as above, forces .
If , then . Replacing by in (2.29) and using similar arguments as above, we find that or .
Hence in both cases we find that The set of for which these two properties hold are additive subgroups of whose union is ; accordingly, we must have either for all , or for all .
Assume that Replacing by with , the last expression becomes Writing instead of we get , that is, In view of Remarks 2.1(5), the above relation yields that for all . Hence, using Remarks 2.1(4), we conclude that contradiction, thus Substitution for in the latter relation, we get Replacing by we obtain so that Now, an application of Remarks 2.1(5) yields that for all which obviously leads to for all . Thus, is centralizing on and from [7, Theorem 1] we get the required result.
Theorem 2.7. Let be a -torsion free prime ring and a nonzero Jordan ideal of . If admits a nonzero derivation such that for all , then is commutative.
3. Differential Identities with Anticommutator
This section is devoted to finding out if commutativity still holds when the commutator in the conditions of the preceding section is replaced by anticommutator.
Theorem 3.1. Let be a -torsion free -prime ring and be a nonzero -Jordan ideal. Then admits no nonzero derivation which commutes with such that for all .
Proof. Assume that there exists a nonzero derivation which commutes with and satisfying Replacing by in (3.1), where , we get Substituting for in (3.2) with , and using (3.2) again, we obtain Replacing by in (3.3) with , and using (3.3) again, we get Writing instead of in (3.4) with , we find that Taking in (3.5), we get and thus Since (3.6) is the same as (2.11), reasoning as in the proof of Theorem 2.3 we conclude that is commutative and (3.1) becomes hence so that and , a contradiction.
Theorem 3.2. Let be a -torsion free prime ring and a nonzero Jordan ideal. Then admits no nonzero derivation such that for all .
Theorem 3.3. Let be a -torsion free -prime ring and a nonzero -Jordan ideal. Then admits no nonzero derivation such that for all .
Proof. Suppose that there exists a derivation such that Replacing by in (3.9), we find that Substituting for in (3.10), where and , we get Replacing by in (3.11), where , and using (3.11) again, we find that Writing instead of in (3.12), where , and using (3.12) we get Taking in (3.13), we obtain , that is, Since (3.14) is the same as (2.29), reasoning as in the proof of Theorem 2.6, it follows that is commutative and thus (3.9) becomes Replacing by in (3.15) where we get Replacing by in (3.16) where we obtain which leads to and therefore .
Theorem 3.4. Let be a -torsion free prime ring and a nonzero Jordan ideal. Then admits no nonzero derivation such that for all .
To end this paper, we give examples proving that our results cannot be extended to semiprime rings.
Example 3.5. Let be a noncommutative semiprime ring, with involution, which admits a nonzero derivation and let . Consider and define a derivation on by setting . Obviously, is a nonzero -Jordan ideal of , where is the involution defined on by . Furthermore, for all ; but is noncommutative. Hence Theorems 2.3, 2.6, 3.1, and 3.3 cannot be extended to a semiprime ring.
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