#### Abstract

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.

#### 1. Introduction

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 [2] 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 [9], 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 [10] 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
and therefore
Let ; from (2.29) it follows, in light of -primeness, that
Assume 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.

As an application of Theorem 2.6, we get the following theorem for which the proof goes through in the same way as the proof of Theorem 2.4.

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.

Using the same arguments as used in the proof of Theorem 2.4, an application of Theorem 3.1 yields the following result.

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 .

Using the same arguments as used in the proof of Theorem 2.4, an application of Theorem 3.3 yields the following result.

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.