Commutativity Theorems for *-Prime Rings with Differential Identities on Jordan Ideals

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, 𝑅 will represent an associative ring with center 𝑍(𝑅). 𝑅 is 2-torsion free if 2𝑥=0 yields 𝑥=0. We recall that 𝑅 is prime if 𝑎𝑅𝑏=0 implies 𝑎=0 or 𝑏=0. A ring with involution (𝑅,∗) is ∗-prime if 𝑎𝑅𝑏=𝑎𝑅𝑏^∗=0 yields 𝑎=0 or 𝑏=0. 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 𝑅^𝑜 denotes the opposite ring of a prime ring 𝑅, then 𝑅×𝑅^𝑜 equipped with the exchange involution ∗_ex, defined by ∗_ex(𝑥,𝑦)=(𝑦,𝑥), is ∗_ex-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 𝑆𝑎_∗(𝑅)={𝑥∈𝑅∣𝑥^∗=𝑥or𝑥^∗=−𝑥} will denote the set of symmetric or skew-symmetric elements of 𝑅. For 𝑥,𝑦∈𝑅,[𝑥,𝑦]=𝑥𝑦−𝑦𝑥 and 𝑥∘𝑦=𝑥𝑦+𝑦𝑥. An additive subgroup 𝐽 of 𝑅 is a Jordan ideal if 𝑥∘𝑟∈𝐽 for all 𝑥∈𝐽 and 𝑟∈𝑅. Moreover, if 𝐽^∗=𝐽, then 𝐽 is called a ∗-Jordan ideal. We will use without explicit mention the fact that if 𝐽 is a Jordan ideal of 𝑅, then 2[𝑅,𝑅]𝐽⊆𝐽 and 2𝐽[𝑅,𝑅]⊆𝐽 [1, Lemma 1]. Moreover, From [2] we have 4𝑗𝑅𝑗⊂𝐽, 4𝑗²𝑅⊂𝐽 and 4𝑅𝑗²⊂𝐽 for all 𝑗∈𝐽.

A mapping 𝑓∶𝑅→𝑅 is called strong commutativity preserving on a subset 𝑆 of 𝑅 if [𝑓(𝑥),𝑓(𝑦)]=[𝑥,𝑦] for all 𝑥,𝑦∈𝑆. An additive mapping 𝑑∶𝑅→𝑅 is called a derivation if 𝑑(𝑥𝑦)=𝑑(𝑥)𝑦+𝑥𝑑(𝑦) holds for all pairs 𝑥,𝑦∈𝑅. 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 𝑅 be a 2-torsion free ∗-prime ring and 𝐽 a nonzero ∗-Jordan ideal.(1) (see [6, Lemma 2]) If 𝑎𝐽𝑏=𝑎^∗𝐽𝑏=0, then 𝑎=0 or 𝑏=0.(2) (see [6, Lemma 3]) If [𝐽,𝐽]=0, then 𝐽⊆𝑍(𝑅).(3) (see [7, Lemma 3]) If 𝐽⊆𝑍(𝑅), then 𝑅 is commutative.(4) (see [9], Lemma 3]) If 𝑑 is a derivation such that 𝑑(𝑥²)=0 for all 𝑥∈𝐽, then 𝑑=0.
We leave the proofs of the following two easy facts to the reader.(5) If 𝑎𝐽𝑎=0, then 𝑎=0. In particular, if 𝑎𝐽=0 or 𝐽𝑎=0, then 𝑎=0.(6) If 𝑅 admits a derivation 𝑑 such that 𝑑²(𝐽)={0}, then 𝑑=0.

Lemma 2.2. Let 𝑅 be a 2-torsion free ∗-prime ring and 𝐽 a nonzero ∗-Jordan ideal. If 𝑅 admits a nonzero derivation 𝑑 such that [𝑑(𝑥),𝑦²]=0 for all 𝑥,𝑦∈𝐽, then 𝑅 is commutative.

Proof. First suppose that 𝑍(𝑅)∩𝐽=0. From [𝑑(4𝑥𝑦²),𝑦²]=0 it follows that 𝑥,𝑦²𝑑𝑦²=0∀𝑥,𝑦∈𝐽. (2.1) Substituting 2[𝑟,𝑠]𝑥 for 𝑥 in (2.1), where 𝑟,𝑠∈𝑅, we obtain [[𝑟,𝑠],𝑦²]𝑥𝑑(𝑦²)=0 which leads to [𝑟,𝑠],𝑦²𝐽𝑑𝑦²=0∀𝑦∈𝐽,𝑟,𝑠∈𝑅. (2.2) For 𝑦∈𝑆𝑎_∗(𝑅)∩𝐽, (2.2) together with Remarks 2.1(1) forces [[𝑟,𝑠],𝑦²]=0, in which case 𝑦²∈𝑍(𝑅), or 𝑑(𝑦²)=0. Since 𝐽∩𝑍(𝑅)=0, in both the cases we arrive at 𝑑𝑦²=0∀𝑦∈𝐽∩𝑆𝑎_∗(𝑅). (2.3) Let 𝑦∈𝐽; using the fact that 𝑑((𝑦+𝑦^∗)²)=0 and 𝑑((𝑦−𝑦^∗)²)=0, we obtain 𝑑(𝑦²)=−𝑑((𝑦^∗)²). Replacing 𝑦 by 𝑦^∗ in (2.2), we get ([[𝑟,𝑠],𝑦²])^∗𝐽𝑑(𝑦²)=0 which combined with (2.2) yields either 𝑑(𝑦²)=0 or 𝑦²∈𝑍(𝑅) so that 𝑦²=0. This implies that 𝑑𝑦²=0∀𝑦∈𝐽, (2.4) and hence by Remarks 2.1(4), 𝑑=0 which contradicts our hypothesis and thus 𝑍(𝑅)∩𝐽≠0.
Now let 0≠𝑢∈𝑍(𝑅)∩𝐽; from [𝑑(4𝑥𝑢²),𝑦²]=0 it follows that [𝑥,𝑦²]𝑑(𝑢²)=0 for all 𝑥,𝑦∈𝐽 and thus either [𝑥,𝑦²]=0 or 𝑑(𝑢²)=0. If [𝑥,𝑦²]=0, then using similar arguments as used in [5, Proof of Theorem  3] we conclude that 𝑅 is commutative. Assume that 𝑑(𝑢²)=0; the fact that [𝑑(4𝑢²𝑟),𝑦²]=0 yields 𝑢²[𝑑(𝑟),𝑦²]=0. Similarly in view of [𝑑(4(𝑢^∗)²𝑟),𝑦²]=0, we find that (𝑢²)^∗𝑅[𝑑(𝑟),𝑦²]=0 and therefore 𝑑(𝑟),𝑦²=0∀𝑦∈𝐽,𝑟∈𝑅. (2.5) Replacing 𝑟 by 𝑑(𝑟)𝑠 in (2.5) and using (2.5) we get 𝑑²(𝑟)[𝑠,𝑦²]=0. Again, replace 𝑠 by 𝑠𝑡 in the last equation, to get 𝑑²(𝑟)𝑠[𝑡,𝑦²]=0 so that 𝑑²(𝑟)𝑅𝑡,𝑦²=0∀𝑦∈𝐽,𝑟,𝑡∈𝑅. (2.6) In view of ∗-primeness, (2.6) assures that either 𝑑²(𝑟)=0 or [𝑡,𝑦²]=0.
If 𝑑²(𝑟)=0 for all 𝑟∈𝑅, then 𝑑=0 by Remarks 2.1(4) contradiction, thus [𝑡,𝑦²]=0 for all 𝑡∈𝑅 and 𝑦∈𝐽, then [5, Proof of Theorem 3] implies that 𝑅 is commutative.

Theorem 2.3. Let 𝑅 be a 2-torsion free ∗-prime ring and 𝐽 a nonzero ∗-Jordan ideal. If 𝑅 admits a nonzero derivation 𝑑, commuting with ∗, such that either 𝑑 is strong commutativity preserving on 𝐽 or [𝑑(𝑥),𝑑(𝑦)]=0 for all 𝑥,𝑦∈𝐽, then 𝑅 is commutative.

Proof. (i) Assume that 𝑑 is strong commutativity preserving on 𝐽. In this case the condition 𝑑≠0 is not necessary. Indeed, if 𝑑=0, then [𝐽,𝐽]=0 and Remarks 2.1(2) forces 𝐽⊆𝑍(𝑅). Thus 𝑅 is commutative by Remarks 2.1(3).
We are given that 𝑑(𝑥),𝑑𝑦=𝑥,𝑦∀𝑥,𝑦∈𝐽. (2.7) Replacing 𝑦 by 4𝑦²𝑧 in the above expression, where 𝑧∈𝐽, we get 𝑑𝑦²[𝑑(𝑥),𝑧]+𝑑(𝑥),𝑦²𝑑(𝑧)=0∀𝑥,𝑦,𝑧∈𝐽. (2.8) Again, replacing 𝑧 by 2[𝑎,𝑢𝑣]𝑧 in (2.8), where 𝑢,𝑣∈𝐽 and 𝑎∈𝑅, and using (2.8) we obtain 𝑑𝑦²[𝑎,𝑢𝑣][𝑑(𝑥),𝑧]+𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑑(𝑧)=0∀𝑢,𝑣,𝑥,𝑦,𝑧∈𝐽,𝑎∈𝑅. (2.9) Putting 4𝑧²𝑡 for 𝑧 in (2.9), where 𝑡∈𝑅 we find that 𝑑𝑦²[𝑎,𝑢𝑣]𝑧²[𝑑(𝑥),𝑡]+𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑑(𝑡)=0 (2.10) for all 𝑢,𝑣,𝑥,𝑦,𝑧∈𝐽 and 𝑎,𝑡∈𝑅. Substituting 𝑡𝑑(𝑥) for 𝑡 in (2.10), then we have [𝑑(𝑥),𝑦²][𝑎,𝑢𝑣]𝑧²𝑡𝑑²(𝑥)=0 and therefore 𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑅𝑑²(𝑥)=0∀𝑢,𝑣,𝑥,𝑦,𝑧∈𝐽,𝑎∈𝑅. (2.11) Let 𝑥₀∈𝐽∩𝑆𝑎_∗(𝑅); from (2.11) it follows that [𝑑(𝑥₀),𝑦²][𝑎,𝑢𝑣]𝑧²𝑅(𝑑²(𝑥₀))^∗=0, so either 𝑑²(𝑥₀)=0 or [𝑑(𝑥₀),𝑦²][𝑎,𝑢𝑣]𝑧²=0.
Suppose that 𝑑𝑥₀,𝑦²[𝑎,𝑢𝑣]𝑧²=0∀𝑢,𝑣,𝑦,𝑧∈𝐽,𝑎∈𝑅. (2.12) Replacing 𝑎 by 𝑎𝑗² in (2.12), where 𝑗∈𝐽, we find that [𝑑(𝑥₀),𝑦²]𝑎[𝑗²,𝑢𝑣]𝑧²=0 and therefore 𝑑𝑥₀,𝑦²𝑅𝑗²,𝑢𝑣𝑧²=0∀𝑗,𝑢,𝑣,𝑦,𝑧∈𝐽. (2.13) Since 𝑥₀∈𝑆𝑎_∗(𝑅), then (2.13) assures that 𝑑𝑥₀,𝑦^2∗𝑅𝑗²,𝑢𝑣𝑧²=0∀𝑗,𝑢,𝑣,𝑦,𝑧∈𝐽. (2.14) In view of (2.13) and (2.14), the ∗-primeness of 𝑅 forces [𝑑(𝑥₀),𝑦²]=0 for all 𝑦∈𝐽 or [𝑗²,𝑢𝑣]𝑧²=0 for all 𝑗,𝑢,𝑣,𝑧∈𝐽.
Suppose that [𝑗²,𝑢𝑣]𝑧²=0 thus 𝑢𝑗²,𝑣𝑧²+𝑗²,𝑢𝑣𝑧²=0∀𝑧,𝑢,𝑣,𝑗∈𝐽. (2.15) Replacing 𝑢 by 2[𝑟,𝑠]𝑢 in (2.15) we arrive at [𝑗²,[𝑟,𝑠]]𝑢𝑣𝑧²=0 and thus [𝑗²,[𝑟,𝑠]]𝑢𝐽𝑧²=0 which leads to [𝑗²,[𝑟,𝑠]]=0 or 𝑧²=0. Hence, in both cases we find that 𝑧²∈𝑍(𝑅) for all 𝑧∈𝐽 and [5, Proof of Theorem 3] assures that 𝑅 is commutative, thereby [𝑑(𝑥₀),𝑦²]=0 for all 𝑦∈𝐽.
In conclusion, 𝑑²𝑥₀=0or𝑑𝑥₀,𝑦²=0∀𝑦∈𝐽∀𝑥₀∈𝐽∩𝑆𝑎_∗(𝑅). (2.16) Let 𝑥∈𝐽; as 𝑥−𝑥^∗∈𝐽∩𝑆𝑎_∗(𝑅), from the above relation it follows that 𝑑²(𝑥)=𝑑²(𝑥^∗)=𝑑²(𝑥)^∗ or [𝑑(𝑥),𝑦²]=[𝑑(𝑥^∗),𝑦²] for all 𝑦∈𝐽.
Assume that 𝑑²(𝑥)=𝑑²(𝑥)^∗. Hence (2.11) can be rewritten as 𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑅𝑑²(𝑥)^∗=0∀𝑢,𝑣,𝑦,𝑧∈𝐽,𝑎∈𝑅. (2.17) Combining the latter relation with (2.11), we get 𝑑²(𝑥)=0 or [𝑑(𝑥),𝑦²][𝑎,𝑢𝑣]𝑧²=0 which makes it possible to conclude, using similar arguments as above, that [𝑑(𝑥),𝑦²]=0 or 𝑑²(𝑥)=0.
Now suppose [𝑑(𝑥),𝑦²]=[𝑑(𝑥^∗),𝑦²]. In relation to (2.11) let 𝑥 be 𝑥^∗; then we have 𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑅𝑑²𝑥^∗=0∀𝑢,𝑣,𝑦,𝑧∈𝐽,𝑎∈𝑅, (2.18) which gives, because 𝑑 commutes with ∗, 𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑅𝑑²(𝑥)^∗=0∀𝑢,𝑣,𝑦,𝑧∈𝐽,𝑎∈𝑅. (2.19) Whence it follows, according to (2.11), that 𝑑²(𝑥)=0 or [𝑑(𝑥),𝑦²]=0. Thus in both cases we find that 𝑑²(𝑥)=0or𝑑(𝑥),𝑦²=0∀𝑥,𝑦∈𝐽. (2.20) Now let 𝐽₁={𝑥∈𝐽∣𝑑²(𝑥)=0} and 𝐽₂={𝑥∈𝐽∣[𝑑(𝑥),𝑦²]=0forall𝑦∈𝐽}. Clearly, 𝐽₁ and 𝐽₂ are additive subgroups of 𝐽 whose union, because of (2.20), is Hence, by Brauer's trick, either 𝐽=𝐽₁ or 𝐽=𝐽₂. If 𝐽=𝐽₁, then 𝑑²(𝐽)=0 and hence, by Remarks 2.1(6), 𝑑=0 thus 𝐽=𝐽₂, so [𝑑(𝑥),𝑦²]=0 for all 𝑥,𝑦∈𝐽, whence it follows, according to Lemma 2.2, that 𝑅 is commutative.
(ii) Assume that 𝑑(𝑥),𝑑𝑦=0∀𝑥,𝑦∈𝐽. (2.21) Substituting 4𝑦²𝑧 for 𝑦 in (2.21), where 𝑧∈𝐽, we get 𝑑𝑦²[𝑑(𝑥),𝑧]+𝑑(𝑥),𝑦²𝑑(𝑧)=0∀𝑥,𝑦,𝑧∈𝐽. (2.22) Since (2.22) is the same as (2.8), reasoning as in the first case, we conclude that 𝑅 is commutative.

In [10] Herstein proved that if 𝑅 is a prime ring of characteristic not 2 equipped with a nonzero derivation 𝑑 such that [𝑑(𝑥),𝑑(𝑦)]=0 for all 𝑥,𝑦∈𝑅, then 𝑅 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 𝑅 be a 2-torsion free prime ring and 𝐽 a nonzero Jordan ideal. If 𝑅 admits a nonzero derivation 𝑑 such that [𝑑(𝑥),𝑑(𝑦)]=0 for all 𝑥,𝑦∈𝐽, then 𝑅 is commutative.

Proof. Let 𝒟 be the additive mapping defined on ℛ=𝑅×𝑅⁰ by 𝒟(𝑥,𝑦)=(𝑑(𝑥),𝑑(𝑦)). Clearly, 𝒟 is a nonzero derivation of ℛ. Moreover, if we set 𝒥=𝐽×𝐽, then 𝒥 is a ∗_ex-Jordan ideal of ℛ such that [𝒟(𝑥,𝑦),𝒟(𝑢,𝑣)]=(0,0) for all (𝑥,𝑦),(𝑢,𝑣)∈𝒥. Since 𝒟 commutes with ∗_ex and ℛ is ∗_ex-prime, then Theorem 2.3 assures that ℛ is commutative and thus so is 𝑅.

An application of similar arguments yields the following.

Theorem 2.5. Let 𝑅 be a 2-torsion free prime ring and 𝐽 a nonzero Jordan ideal. If 𝑅 admits a derivation 𝑑 strong commutativity preserving on 𝐽, then 𝑅 is commutative.

In 2010, Oukhtite et al. [8, Theorem 2] established that if a 2-torsion free ∗-prime ring admits a nonzero derivation 𝑑 such that [𝑑(𝑥),𝑑(𝑦)]=𝑑([𝑥,𝑦]) for all 𝑥,𝑦 in a nonzero square closed Lie ideal 𝑈, then 𝑑=0 or 𝑈⊂𝑍(𝑅). Motivated by this result, our aim in the following theorem is to explore the commutativity of ∗-prime rings admitting a nonzero derivation 𝑑 satisfying the above condition on ∗-Jordan ideals.

Theorem 2.6. Let 𝑅 be a 2-torsion free ∗-prime ring and 𝐽 a nonzero ∗-Jordan ideal. If 𝑅 admits a nonzero derivation 𝑑 which commutes with ∗ such that [𝑑(𝑥),𝑑(𝑦)]=𝑑([𝑥,𝑦]) for all 𝑥,𝑦∈𝐽, then 𝑅 is commutative.

Proof. Suppose that 𝑑(𝑥),𝑑𝑦=𝑑𝑥,𝑦∀𝑥,𝑦∈𝐽. (2.23) Replacing 𝑥 by 2[𝑎,𝑢𝑣]𝑥 in (2.23) where 𝑢,𝑣∈𝐽 and 𝑎∈𝑅, in light of 2[𝑎,𝑢𝑣]∈𝐽, we find that 𝑑([𝑎,𝑢𝑣])𝑥,𝑑𝑦−𝑦+[𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑑(𝑥)=0∀𝑢,𝑣,𝑥,𝑦∈𝐽,𝑎∈𝑅. (2.24) Again replacing 𝑥 by 2[𝑟,𝑝𝑞]𝑥 in (2.24) with 𝑝,𝑞∈𝐽 and 𝑟∈𝑅, we get 𝑑([𝑎,𝑢𝑣])𝑟,𝑝𝑞𝑥,𝑑𝑦−𝑦+[𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑑(𝑥)=0. (2.25) Substituting 𝑥 by 4𝑥²𝑡 in (2.25) with 𝑡∈𝑅 and employing (2.25), we obtain 𝑑([𝑎,𝑢𝑣])𝑟,𝑝𝑞𝑥²𝑡,𝑑𝑦−𝑦+[𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑥²𝑑(𝑡)=0. (2.26) Replacing 𝑡 by 𝑡𝑠 in (2.26), where 𝑠∈𝑅, then we have 𝑑([𝑎,𝑢𝑣])𝑟,𝑝𝑞𝑥²𝑡𝑠,𝑑𝑦−𝑦+[𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑥²𝑡𝑑(𝑠)=0, (2.27) for all 𝑝,𝑞,𝑢,𝑣,𝑥,𝑦∈𝐽 and for all 𝑎,𝑟,𝑡,𝑠∈𝑅. Putting 𝑠=𝑑(𝑦)−𝑦 in (2.27), we find that [𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑥²𝑡𝑑𝑑𝑦−𝑦=0 (2.28) and therefore [𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑥²𝑅𝑑𝑑𝑦−𝑦=0∀𝑝,𝑞,𝑢,𝑣,𝑥,𝑦∈𝐽,𝑎,𝑟∈𝑅. (2.29) Let 𝑦∈𝐽∩𝑆𝑎_∗(𝑅); from (2.29) it follows, in light of ∗-primeness, that [𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑥²=0or𝑑𝑑𝑦−𝑦=0. (2.30) Assume that [𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑥²=0∀𝑝,𝑞,𝑢,𝑣,𝑥∈𝐽,𝑎,𝑟∈𝑅. (2.31) Replacing 𝑟 by 𝑟𝑗² in the above expression, we get [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦]𝑟[𝑗²,𝑝𝑞]𝑥²=0 and thus [𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑅𝑗²,𝑝𝑞𝑥²=0. (2.32)𝑅 being ∗-prime implies that either [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦]=0 or [𝑗²,𝑝𝑞]𝑥²=0.
If [𝑗²,𝑝𝑞]𝑥²=0, then 𝑝𝑗²,𝑞𝑥²+𝑗²,𝑝𝑞𝑥²=0∀𝑗,𝑝,𝑞,𝑥∈𝐽. (2.33) As (2.33) is the same as (2.15), then reasoning as in the proof of Theorem 2.3 we find that 𝑅 is commutative and therefore [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦]=0.
In conclusion, [𝑎,𝑢𝑣],𝑑𝑦−𝑦=0or𝑑𝑑𝑦−𝑦=0∀𝑦∈𝐽∩𝑆𝑎_∗. (2.34) Let 𝑦∈𝐽; as 𝑦−𝑦^∗∈𝐽∩𝑆𝑎_∗(𝑅), in view of (2.34) we obtain that either 𝑑(𝑑(𝑦−𝑦^∗)−(𝑦−𝑦^∗))=0 or [[𝑎,𝑢𝑣],𝑑(𝑦−𝑦^∗)−(𝑦−𝑦^∗)]=0.
If 𝑑(𝑑(𝑦−𝑦^∗)−(𝑦−𝑦^∗))=0, then 𝑑(𝑑(𝑦)−𝑦)=𝑑(𝑑(𝑦^∗)−𝑦^∗) and hence, because 𝑑 commutes with ∗, (2.29) reduces to [𝑎,𝑢𝑣],𝑑𝑦−𝑦𝑟,𝑝𝑞𝑥²𝑅𝑑𝑑𝑦−𝑦^∗=0. (2.35) In light of (2.29), the latter expression together with ∗-primeness of 𝑅 shows that either 𝑑(𝑑(𝑦)−𝑦)=0 or [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦][𝑟,𝑝𝑞]𝑥²=0 which, as above, forces [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦]=0.
If [[𝑎,𝑢𝑣],𝑑(𝑦−𝑦^∗)−(𝑦−𝑦^∗)]=0, then [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦]=[[𝑎,𝑢𝑣],𝑑(𝑦^∗)−𝑦^∗]. Replacing 𝑦 by 𝑦^∗ in (2.29) and using similar arguments as above, we find that [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦]=0 or 𝑑(𝑑(𝑦)−𝑦)=0.
Hence in both cases we find that [𝑎,𝑢𝑣],𝑑𝑦−𝑦=0or𝑑𝑑𝑦−𝑦=0∀𝑦,𝑢,𝑣∈𝐽,𝑎∈𝑅. (2.36) The set of 𝑦∈𝐽 for which these two properties hold are additive subgroups of 𝐽 whose union is 𝐽; accordingly, we must have either [[𝑎,𝑢𝑣],𝑑(𝑦)−𝑦]=0 for all 𝑢,𝑣,𝑦∈𝐽, 𝑎∈𝑅 or 𝑑(𝑑(𝑦)−𝑦)=0 for all 𝑦∈𝐽.
Assume that 𝑑²𝑦=𝑑𝑦∀𝑦∈𝐽. (2.37) Replacing 𝑦 by 4𝑦²𝑧 with 𝑧∈𝐽, the last expression becomes 𝑑𝑦²𝑑(𝑧)=0. (2.38) Writing 4𝑧𝑦² instead of 𝑧 we get 𝑑(𝑦²)𝑧𝑑(𝑦²)=0, that is, 𝑑𝑦²𝐽𝑑𝑦²=0∀𝑦∈𝐽. (2.39) In view of Remarks 2.1(5), the above relation yields that 𝑑(𝑦²)=0 for all 𝑦∈𝐽. Hence, using Remarks 2.1(4), we conclude that 𝑑=0 contradiction, thus [𝑎,𝑢𝑣],𝑑𝑦−𝑦=0∀𝑢,𝑣,𝑦∈𝐽,𝑎∈𝑅. (2.40) Substitution [𝑟,𝑠](𝑑(𝑦)−𝑦) for 𝑎 in the latter relation, we get [𝑟,𝑠],𝑑𝑦−𝑦𝑢𝑑𝑦−𝑦,𝑣+[𝑟,𝑠],𝑑𝑦−𝑦𝑑𝑦−𝑦,𝑢𝑣=0. (2.41) Replacing 𝑣 by 2𝑣[𝑟,𝑠] we obtain [𝑟,𝑠],𝑑𝑦−𝑦𝑢𝑣𝑑𝑦−𝑦,[𝑟,𝑠]=0 (2.42) so that 𝑣[𝑟,𝑠],𝑑𝑦−𝑦𝐽𝑣[𝑟,𝑠],𝑑𝑦−𝑦=0∀𝑣,𝑦∈𝐽and𝑟,𝑠∈𝑅. (2.43) Now, an application of Remarks 2.1(5) yields that [[𝑟,𝑠],𝑑(𝑦)−𝑦]=0 for all 𝑦∈𝐽,𝑟,𝑠∈𝑅 which obviously leads to 𝑑(𝑦)−𝑦∈𝑍(𝑅) for all 𝑦∈𝐽. Thus, 𝑑 is centralizing on 𝐽 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 𝑅 be a 2-torsion free prime ring and 𝐽 a nonzero Jordan ideal of 𝑅. If 𝑅 admits a nonzero derivation 𝑑 such that [𝑑(𝑥),𝑑(𝑦)]=𝑑([𝑥,𝑦]) for all 𝑥,𝑦∈𝐽, then 𝑅 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 𝑅 be a 2-torsion free ∗-prime ring and 𝐽 be a nonzero ∗-Jordan ideal. Then 𝑅 admits no nonzero derivation 𝑑 which commutes with ∗ such that 𝑑(𝑥)∘𝑑(𝑦)=0 for all 𝑥,𝑦∈𝐽.

Proof. Assume that there exists a nonzero derivation 𝑑 which commutes with ∗ and satisfying 𝑑(𝑥)∘𝑑𝑦=0∀𝑥,𝑦∈𝐽. (3.1) Replacing 𝑦 by 4𝑦²𝑧 in (3.1), where 𝑧∈𝐽, we get −𝑑𝑦²[𝑑(𝑥),𝑧]+𝑑(𝑥),𝑦²𝑑(𝑧)=0. (3.2) Substituting 2[𝑎,𝑢𝑣]𝑧 for 𝑧 in (3.2) with 𝑢,𝑣∈𝐽, 𝑎∈𝑅 and using (3.2) again, we obtain −𝑑𝑦²[𝑎,𝑢𝑣][𝑑(𝑥),𝑧]+𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑑(𝑧)=0. (3.3) Replacing 𝑧 by 4𝑧²𝑡 in (3.3) with 𝑡∈𝑅, and using (3.3) again, we get −𝑑𝑦²[𝑎,𝑢𝑣]𝑧²[𝑑(𝑥),𝑡]+𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑑(𝑡)=0. (3.4) Writing 𝑡𝑤 instead of 𝑡 in (3.4) with 𝑤∈𝑅, we find that −𝑑𝑦²[𝑎,𝑢𝑣]𝑧²𝑡[𝑑(𝑥),𝑤]+𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑡𝑑(𝑤)=0. (3.5) Taking 𝑤=𝑑(𝑥) in (3.5), we get [𝑑(𝑥),𝑦²][𝑎,𝑢𝑣]𝑧²𝑡𝑑²(𝑥)=0 and thus 𝑑(𝑥),𝑦²[𝑎,𝑢𝑣]𝑧²𝑅𝑑²(𝑥)=0. (3.6) Since (3.6) is the same as (2.11), reasoning as in the proof of Theorem 2.3 we conclude that 𝑅 is commutative and (3.1) becomes 2𝑑(𝑥)𝑑𝑦=0∀𝑥,𝑦∈𝐽 (3.7) hence 𝑑(𝑥)𝑅𝑑𝑦=0∀𝑥,𝑦∈𝐽 (3.8) so that 𝑑(𝐽)=0 and 𝑑=0, 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 𝑅 be a 2-torsion free prime ring and 𝐽 a nonzero Jordan ideal. Then 𝑅 admits no nonzero derivation 𝑑 such that 𝑑(𝑥)∘𝑑(𝑦)=0 for all 𝑥,𝑦∈𝐽.

Theorem 3.3. Let 𝑅 be a 2-torsion free ∗-prime ring and 𝐽 a nonzero ∗-Jordan ideal. Then 𝑅 admits no nonzero derivation 𝑑 such that 𝑑(𝑥)∘𝑑(𝑦)=𝑑(𝑥∘𝑦) for all 𝑥,𝑦∈𝐽.

Proof. Suppose that there exists a derivation 𝑑 such that 𝑑(𝑥)∘𝑑𝑦=𝑑𝑥∘𝑦∀𝑥,𝑦∈𝐽. (3.9) Replacing 𝑦 by 2𝑦[𝑟,𝑠] in (3.9), we find that −𝑑𝑦[𝑑(𝑥)−𝑥,[𝑟,𝑠]]+(𝑑(𝑥)−𝑥)∘𝑦𝑑([𝑟,𝑠])−𝑦[𝑑(𝑥),𝑑([𝑟,𝑠])]+𝑦𝑑([𝑥,[𝑟,𝑠]])=0. (3.10) Substituting 2𝑦[𝑎,𝑢𝑣] for 𝑦 in (3.10), where 𝑢,𝑣∈𝐽 and 𝑎∈𝑅, we get −𝑑𝑦[𝑎,𝑢𝑣][𝑑(𝑥)−𝑥,[𝑟,𝑠]]+𝑑(𝑥)−𝑥,𝑦[𝑎,𝑢𝑣]𝑑([𝑟,𝑠])=0. (3.11) Replacing 𝑦 by 4𝑡𝑦² in (3.11), where 𝑡∈𝑅, and using (3.11) again, we find that −𝑑(𝑡)𝑦²[𝑎,𝑢𝑣][𝑑(𝑥)−𝑥,[𝑟,𝑠]]+[𝑑(𝑥)−𝑥,𝑡]𝑦²[𝑎,𝑢𝑣]𝑑([𝑟,𝑠])=0. (3.12) Writing 𝑤𝑡 instead of 𝑡 in (3.12), where 𝑤∈𝑅, and using (3.12) we get −𝑑(𝑤)𝑡𝑦²[𝑎,𝑢𝑣][𝑑(𝑥)−𝑥,[𝑟,𝑠]]+[𝑑(𝑥)−𝑥,𝑤]𝑡𝑦²[𝑎,𝑢𝑣]𝑑([𝑟,𝑠])=0. (3.13) Taking 𝑤=𝑑(𝑥)−𝑥 in (3.13), we obtain 𝑑(𝑑(𝑥)−𝑥)𝑡𝑦²[𝑎,𝑢𝑣][𝑑(𝑥)−𝑥,[𝑟,𝑠]]=0, that is, 𝑑(𝑑(𝑥)−𝑥)𝑅𝑦²[𝑎,𝑢𝑣][𝑑(𝑥)−𝑥,[𝑟,𝑠]]=0. (3.14) Since (3.14) is the same as (2.29), reasoning as in the proof of Theorem 2.6, it follows that 𝑅 is commutative and thus (3.9) becomes 2𝑑(𝑥)𝑑𝑦−2𝑑(𝑥)𝑦−2𝑥𝑑𝑦=0. (3.15) Replacing 𝑦 by 2𝑦𝑧 in (3.15) where 𝑧∈𝐽 we get 𝑑(𝑥)𝑦𝑑(𝑧)−𝑥𝑦𝑑(𝑧)=0. (3.16) Replacing 𝑥 by 2𝑥𝑡 in (3.16) where 𝑡∈𝐽 we obtain 𝑑(𝑡)𝑥𝑦𝑑(𝑧)=0 (3.17) which leads to 𝑑(𝐽)=0 and therefore 𝑑=0.

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 𝑅 be a 2-torsion free prime ring and 𝐽 a nonzero Jordan ideal. Then 𝑅 admits no nonzero derivation 𝑑 such that 𝑑(𝑥)∘𝑑(𝑦)=𝑑(𝑥∘𝑦) for all 𝑥,𝑦∈𝐽.

To end this paper, we give examples proving that our results cannot be extended to semiprime rings.

Example 3.5. Let (𝑅₁,∗) be a noncommutative semiprime ring, with involution, which admits a nonzero derivation 𝑑 and let 𝑅=𝑅₁×𝑅₁. Consider 𝐽=0×𝑅₁ and define a derivation 𝐷 on 𝑅 by setting 𝐷(𝑥,𝑦)=(𝑑(𝑥),0). Obviously, 𝐽 is a nonzero 𝜏-Jordan ideal of 𝑅, where 𝜏 is the involution defined on 𝑅 by 𝜏(𝑥,𝑦)=(𝑥^∗,𝑦^∗). Furthermore, [𝐷(𝑢)(𝑣[𝐷(𝑢)(𝑣,𝐷(𝑢)(𝑣)(𝑢)(𝑣)(),𝐷)]=0,,𝐷)]=𝐷([𝑢,𝑣])∘𝐷=0,𝐷∘𝐷=𝐷𝑢∘𝑣 (3.18) for all 𝑢,𝑣∈𝐽; but 𝑅 is noncommutative. Hence Theorems 2.3, 2.6, 3.1, and 3.3 cannot be extended to a semiprime ring.