Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

Abstract

Let 𝑅 be a 2-torsion free ring and let 𝐿 be a noncentral Lie ideal of 𝑅, and let 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 be two generalized derivations of 𝑅. We will analyse the structure of 𝑅 in the following cases: (a) 𝑅 is prime and 𝐹(𝑢^𝑚)=𝐺(𝑢^𝑛) for all 𝑢∈𝐿 and fixed positive integers 𝑚≠𝑛; (b) 𝑅 is prime and 𝐹((𝑢^𝑝𝑣^𝑞)^𝑚)=𝐺((𝑣^𝑟𝑢^𝑠)^𝑛) for all 𝑢,𝑣∈𝐿 and fixed integers 𝑚,𝑛,𝑝,𝑞,𝑟,𝑠≥1; (c) 𝑅 is semiprime and 𝐹((𝑢𝑣)^𝑛)=𝐺((𝑣𝑢)^𝑛) for all 𝑢,𝑣∈[𝑅,𝑅] and fixed integer 𝑛≥1; and (d) 𝑅 is semiprime and 𝐹((𝑢𝑣)^𝑛)=𝐺((𝑣𝑢)^𝑛) for all 𝑢,𝑣∈𝑅 and fixed integer 𝑛≥1.

1. Introduction

Let 𝑅 be an associative ring with characteristic different from 2, 𝑍(𝑅) its center, 𝑈 its (right) Utumi quotient ring, and 𝐶 its extended centroid. The simple commutator 𝑎𝑏−𝑏𝑎 will be denoted by [𝑎,𝑏]. Recall that a derivation 𝑑∶𝑅→𝑅 is an additive map satisfying the product rule 𝑑(𝑥𝑦)=𝑑(𝑥)𝑦+𝑥𝑑(𝑦) for all 𝑥,𝑦∈𝑅.

A left multiplier 𝐻 on a ring 𝑅 is an additive map satisfying the rule 𝐻(𝑥𝑦)=𝐻(𝑥)𝑦 for all 𝑥,𝑦∈𝑅. In case there exists an endomorphism 𝑔 of 𝑅 such that 𝐻(𝑥𝑦)=𝐻(𝑥)𝑔(𝑦) for all 𝑥,𝑦∈𝑅, then 𝐻 is called left 𝑔-multiplier of 𝑅.

A generalized derivation 𝐹 on a ring 𝑅 is an additive map satisfying 𝐹(𝑥𝑦)=𝐹(𝑥)𝑦+𝑥𝑑(𝑦) for all 𝑥,𝑦∈𝑅 and some derivation 𝑑 of 𝑅. A significative example is a map of the form 𝐹(𝑥)=𝑎𝑥+𝑥𝑏, for some 𝑎,𝑏∈𝑅; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from the algebraic point of view might be interesting (see, e.g., [1]). Notice that any derivation is a generalized one and also that the generalized inner derivations include left multipliers and right multipliers. Thus the concept of generalized derivation covers both the concept of derivation and the concept of left (right) multipliers.

Since the sum of two generalized derivations is a generalized derivation, of course every map of the form 𝐹(𝑥)=𝑐𝑥+𝑑(𝑥) is a generalized derivation on 𝑅, where 𝑐 is a fixed element of 𝑅 and 𝑑 is a derivation of 𝑅.

In [1, Theorem 3] Lee proved that every generalized derivation 𝐹 on a dense right ideal of 𝑅 can be uniquely extended to the Utumi quotient ring 𝑈 of 𝑅, and thus any generalized derivation of 𝑅 can be defined on the whole 𝑈; moreover it is of the form 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥) for some 𝑎∈𝑈 and 𝑑 is a derivation on 𝑈 (𝐹 is said to be a generalized derivation associated with derivation 𝑑).

Many results in the literature indicate that the global structure of a ring 𝑅 is often tightly connected to the behaviour of additive mappings defined on 𝑅.

In [2] Bergen proved that if 𝑔 is an automorphism of 𝑅 such that (𝑔(𝑥)−𝑥)^𝑚=0, for all 𝑥∈𝑅, where 𝑚≥1 is a fixed integer, then 𝑔=1. Daif and Bell [3] showed some results which have the same flavour, when the automorphism is replaced by a nonzero derivation 𝑑. In [3] it is proved that if 𝑅 is a semiprime ring with a nonzero ideal 𝐼 such that 𝑑([𝑥,𝑦])−[𝑥,𝑦]=0, or 𝑑([𝑥,𝑦])+[𝑥,𝑦]=0, for all 𝑥,𝑦∈𝐼, then 𝐼 is central. Later Hongan [4] proved that if 𝑅 is a 2-torsion free semiprime ring and 𝐼 a nonzero ideal of 𝑅, then 𝐼 is central if and only if 𝑑([𝑥,𝑦])−[𝑥,𝑦]∈𝑍(𝑅), or 𝑑([𝑥,𝑦])+[𝑥,𝑦]∈𝑍(𝑅), for all 𝑥,𝑦∈𝐼. Recently in [5] Ashraf and Ali obtained commutativity theorems for prime rings admitting left multipliers which satisfy similar conditions. More precisely in [5] it is showed that a prime ring 𝑅 must be commutative if there exist a nonzero ideal 𝐼 of 𝑅 and a left multiplier 𝐻, which is not the identity map on 𝐼, such that one of the following holds: (i) 𝐻([𝑥,𝑦])=[𝑥,𝑦] for all 𝑥,𝑦∈𝐼 [5, Theorem 2.1]; (ii) 𝐻(𝑥𝑦)−𝑥𝑦∈𝑍(𝑅) for all 𝑥,𝑦∈𝐼 [5, Theorem 3.1]; and (iii) 𝐻(𝑥𝑦)−𝑦𝑥∈𝑍(𝑅) for all 𝑥,𝑦∈𝐼 [5, Theorem 3.3]. Moreover the same results hold in case 𝐻 is replaced by a generalized derivation 𝐹 of 𝑅.

In a more recent paper [6], Ali and Huang extended the previous cited results in the case 𝑅 is a semiprime ring, 𝐼 is a nonzero ideal of 𝑅, and 𝐻 is a left 𝑔-multiplier of 𝑅. They proved that 𝐼⊆𝑍(𝑅) if one of the following holds: (i) 𝐻([𝑥,𝑦])=[𝑥,𝑦] for all 𝑥,𝑦∈𝐼 [6, Theorem 2.1]; (ii) 𝐻(𝑥𝑦)=𝑥𝑦 for all 𝑥,𝑦∈𝐼 [6, Theorem 2.2]; (iii) 𝐻(𝑥𝑦)=𝑦𝑥 for all 𝑥,𝑦∈𝐼 [6, Theorem 2.3]; and (iv) 𝐻(𝑥²)=𝑥², for all 𝑥∈𝐼 [6, Theorem 2.5]. Moreover, in case 𝑅 is a prime ring, the same conditions force the commutativity of 𝑅.

A natural question is to consider additive maps 𝐹,𝐺∶𝑅→𝑅 such that 𝐹(𝑓(𝑥,𝑦))=𝐺(𝑔(𝑥,𝑦)), when 𝑓(𝑥,𝑦) and 𝑔(𝑥,𝑦) are both either monomials or powers of the commutator [𝑥,𝑦]. In this sense, in [7] it is proved that, under appropriate torsion assumptions, a prime ring 𝑅 is commutative if it admits a nonzero derivation 𝑑 satisfying one of the following: (i) 𝑑(𝑥^𝑛𝑦^𝑚)=𝑑(𝑦^𝑚𝑥^𝑛); (ii) 𝑑((𝑥𝑦)^𝑛)=𝑑((𝑦𝑥)^𝑛); (iii) 𝑑((𝑥𝑦)^𝑛)=𝑑(𝑥^𝑛𝑦^𝑛); and (iv) 𝑑((𝑥𝑦)^𝑛)=𝑑(𝑦^𝑛𝑥^𝑛), for all 𝑥,𝑦∈𝑅.

So it seems natural to ask about the case when the derivation 𝑑 is replaced by a generalized derivation. Motivated by the previous cited results, in this paper we will introduce two different generalized derivations acting on 𝑅 and satisfying some appropriate conditions on some suitable subsets of 𝑅. We will prove the following.

Theorem 1. Let 𝑅 be a 2-torsion free prime ring and let 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 be two generalized derivations associated with derivations 𝑑∶𝑅→𝑅 and 𝛿∶𝑅→𝑅, respectively. Suppose that there exist 𝑚,𝑛≥1 integers such that 𝐹([𝑥,𝑦]^𝑚)=𝐺([𝑥,𝑦]^𝑛) for all 𝑥,𝑦∈𝑅. Then either 𝑅 satisfies the standard identity 𝑠₄ or one of the following holds:(a)𝐹=𝐺 is an inner ordinary derivation of 𝑅 and for all 𝑥₁,𝑥₂∈𝑅 there exists 𝛼∈𝐶 such that [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛+𝛼;(b)𝐹=𝐺 and [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛, for all 𝑥₁,𝑥₂∈𝑅.

Theorem 2. Let 𝑅 be a 2-torsion free prime ring and let 𝐿 be a noncentral Lie ideal of 𝑅, and let 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 be two nonzero generalized derivations associated with derivations 𝑑∶𝑅→𝑅 and 𝛿∶𝑅→𝑅, respectively. Suppose that there exist 𝑚,𝑛,𝑝,𝑞,𝑟,𝑠≥1 integers such that 𝐹((𝑢^𝑝𝑣^𝑞)^𝑚)=𝐺((𝑣^𝑟𝑢^𝑠)^𝑛) for all 𝑢,𝑣∈𝐿. Then 𝑅 satisfies the standard identity 𝑠₄.

In the last section we study some commutativity conditions for a semiprime ring with a generalized derivation satisfying suitable algebraic conditions. More precisely, we will prove the following.

Theorem 3. Let 𝑅 be a 2-torsion free semiprime ring and let 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 be generalized derivations associated, respectively, with derivations 𝑑∶𝑅→𝑅 and 𝛿∶𝑅→𝑅. Set 𝐿=[𝑅,𝑅] and 𝑛≥1 such that 𝐹((𝑢𝑣)^𝑛)=𝐺((𝑣𝑢)^𝑛) for all 𝑢,𝑣∈𝐿. Then there exists a central idempotent 𝑒 of  𝑈 such that, on the direct sum decomposition 𝑈=𝑒𝑈⊕(1−𝑒)𝑈, the generalized derivations 𝐹 and 𝐺 vanish identically on 𝑒𝑈 and the ring (1−𝑒)𝑈 satisfies 𝑠₄.

Theorem 4. Let 𝑅 be a 2-torsion free semiprime ring and let 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 be generalized derivations associated with derivations 𝑑∶𝑅→𝑅 and 𝛿∶𝑅→𝑅, respectively. Set 𝑛≥1 such that 𝐹((𝑢𝑣)^𝑛)=𝐺((𝑣𝑢)^𝑛) for all 𝑢,𝑣∈𝑅. Then 𝑅 contains a nonzero central ideal, 𝑑(𝑅)⊆𝑍(𝑅), 𝛿(𝑅)⊆𝑍(𝑅), and there exist 𝜆,𝜇∈𝐶 such that 𝐹(𝑥)=𝜆𝑥+𝑑(𝑥), 𝐺(𝑥)=𝜇𝑥+𝛿(𝑥), for all 𝑥∈𝑅.
Moreover there exists a central idempotent 𝑒 of  𝑈 such that, on the direct sum decomposition 𝑈=𝑒𝑈⊕(1−𝑒)𝑈, the generalized derivations 𝐹 and 𝐺 vanish identically on 𝑒𝑈 and the ring (1−𝑒)𝑈 is commutative.

2. Action of Generalized Derivations on Prime Rings

In order to prove the main result in this section, we first fix some intermediate lemmas and theorems. We begin with the following.

Remark 5 (see [8], Lemma 3). Let 𝑅 be a prime ring with extended centroid 𝐶 and let 𝑓(𝑥₁,…,𝑥_𝑛) be any polynomial, not necessarily multilinear. Then the right (left) annihilator of 𝑓(𝑥₁,…,𝑥_𝑛) is zero, unless when 𝑓(𝑥₁,…,𝑥_𝑛) is an identity for 𝑅.

Lemma 6. Let 𝑅 be a primitive ring which is isomorphic to a dense ring of linear transformations of a vector space 𝑉 over the field 𝐶 such that dim_𝐶𝑉≥2 and let 𝑎,𝑏∈𝑅, 𝑚,𝑛≥1, different integers. If for any 𝑥,𝑦∈𝑅, [𝑥,𝑦]^𝑚𝑎=[𝑥,𝑦]^𝑛𝑏, then 𝑎=𝑏 and [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛 for all 𝑥₁,𝑥₂∈𝑅, unless when 𝑎=𝑏=0.

Proof. Let 0≠𝑣∈𝑉. Since dim_𝐶𝑉≥2, there exists 𝑤∈𝑉 such that {𝑣,𝑤} are linearly 𝐶-independent. By the density of 𝑅 there exist 𝑥,𝑦∈𝑅 such that 𝑣𝑥=𝑤,𝑣𝑦=0,𝑤𝑦=𝑣 (1) so that 𝑣[𝑥,𝑦]=𝑣 and by the hypothesis 0=𝑣𝑥,𝑦^𝑚𝑎−𝑥,𝑦^𝑛𝑏=𝑣(𝑎−𝑏) (2) that is 𝑉(𝑎−𝑏)=(0), which implies 𝑎=𝑏. Hence ([𝑥₁,𝑥₂]^𝑚−[𝑥₁,𝑥₂]^𝑛)𝑎=0 for all 𝑥₁,𝑥₂∈𝑅 and by Remark 5 we have that either 𝑎=𝑏=0 or [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛 for any 𝑥₁,𝑥₂∈𝑅.

Lemma 7. Let 𝑅 be a 2-torsion free prime ring, 𝐽 a nonzero two-sided ideal of 𝑅, and 𝑓(𝑥₁,…,𝑥_𝑛) a noncentral polynomial over 𝐶, the extended centroid of 𝑅. If 𝐹∶𝑅→𝑅 is a generalized derivation associated with a derivation 𝑑∶𝑅→𝑅 such that 𝐹(𝑓(𝑟₁,…,𝑟_𝑛))∈𝐶 for all 𝑟₁,…,𝑟_𝑛∈𝐽, then 𝐹(𝑥)=0, for all 𝑥∈𝑅.

Proof. In light of previous remarks, we have that there exists 𝑎∈𝑈, the Utumi quotient ring of 𝑅, such that 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥), for all 𝑥∈𝑅. Thus 𝐽 satisfies the generalized differential identity 𝑎𝑓𝑥₁,…,𝑥_𝑛+𝑑𝑓𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1. (3)
Since, by [9], 𝐽 and 𝑅 satisfy the same differential identities, then we have that 𝑅 satisfies 𝐹(𝑓(𝑥₁,…,𝑥_𝑛))∈𝐶. Let 𝑆 be the additive subgroup generated by the subset 𝑓𝑟₁,…,𝑟_𝑛∣𝑟₁,…,𝑟_𝑛∈𝑅. (4)
𝑆 is a Lie ideal of 𝑅; indeed for any 𝑟∈𝑅, 𝑦₁,…,𝑦_𝑛∈𝑅 one has 𝑟,𝑓𝑦₁,…,𝑦_𝑛=_𝑖𝑓𝑦₁,…,𝑟,𝑦_𝑖,…,𝑦_𝑛∈𝑆. (5)
If 𝑆 is noncommutative then, by [10, pages 4-5], there exists a nonzero two-sided ideal 𝐼 of 𝑅 such that 0≠[𝐼,𝑅]⊆𝑆. In this case it is easy to see that 𝐹([𝑟₁,𝑟₂])∈𝐶, for all 𝑟₁,𝑟₂∈𝐼. In particular [𝐹(𝑢),𝑢]=0 for all 𝑢∈[𝐼,𝐼]. Since [𝐼,𝐼] is a noncentral Lie ideal of 𝑅, it follows easily that 𝐹 must be zero (see, e.g., Theorem 3.3 in [11]).
Hence we may consider it the only case when 𝑆 is commutative.
Thus [𝑓(𝑥₁,…,𝑥_𝑛),𝑓(𝑦₁,…,𝑦_𝑛)] is an identity in 𝑅. This means that there exist a field 𝐾 and a positive integer 𝑚 such that [𝑓(𝑥₁,…,𝑥_𝑛),𝑓(𝑦₁,…,𝑦_𝑛)] is also an identity in 𝑀_𝑚(𝐾). If 𝑚=1, 𝑅 is commutative, thus we suppose 𝑚≥2.
Since 𝑓(𝑥₁,…,𝑥_𝑛) is not central valued on 𝑅, there exist 𝑟₁,…,𝑟_𝑛∈𝑀_𝑚(𝐾) such that 𝑓(𝑟₁,…,𝑟_𝑛)=𝑎∉𝑍(𝑅), so that [𝑎,𝑓(𝑦₁,…,𝑦_𝑛)] is also a generalized identity in 𝑀_𝑚(𝐾). By a result of Lee (see [12], Theorem), we have the contradiction that 𝑓(𝑥₁,…,𝑥_𝑛) is central valued on 𝑅.

Lemma 8. Let 𝑅 be a 2-torsion free primitive ring which is isomorphic to a dense ring of linear transformations of a vector space 𝑉 over the field 𝐶 such that dim_𝐶𝑉≥3, and let 𝑎,𝑏∈𝑅, 𝛾∈𝐶, 𝑚,𝑛≥1, different integers. If for any 𝑥,𝑦∈𝑅, 𝛾[𝑥,𝑦]^𝑚+𝑎[𝑥,𝑦]^𝑚+[𝑥,𝑦]^𝑚𝑏=𝑎[𝑥,𝑦]^𝑛+[𝑥,𝑦]^𝑛𝑏, then 𝛾=0, 𝑎+𝑏=0 and one of the following holds:(a)𝑎,𝑏∈𝐶;(b)for all 𝑥₁,𝑥₂∈𝑅 there exists 𝛼∈𝐶 such that [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛+𝛼.

Proof. Let 0≠𝑣∈𝑉 such that {𝑣,𝑣𝑎} are linearly 𝐶-independent. Since dim_𝐶𝑉≥3, there exists 𝑤∈𝑉 such that {𝑣,𝑣𝑎,𝑤} are linearly 𝐶-independent. By the density of 𝑅 there exist 𝑥,𝑦∈𝑅 such that 𝑣𝑎𝑥=𝑤,𝑣𝑎𝑦=0,𝑣𝑥=𝑤,𝑣𝑦=0,𝑤𝑦=𝑣 (6) so that 𝑣[𝑥,𝑦]=𝑣, 𝑣𝑎[𝑥,𝑦]=0 and by the hypothesis 0=𝑣𝛾𝑥,𝑦^𝑚+𝑎𝑥,𝑦^𝑚+𝑥,𝑦^𝑚𝑏−𝑎𝑥,𝑦^𝑛−𝑥,𝑦^𝑛𝑏=𝛾𝑣 (7) which implies 𝛾=0. Thus 𝑎([𝑥,𝑦]^𝑚−[𝑥,𝑦]^𝑛)+([𝑥,𝑦]^𝑚−[𝑥,𝑦]^𝑛)𝑏=0 for all 𝑥,𝑦∈𝑅 and by Lemma 7 we get the required conclusions.
Assume now that, for any 𝑣∈𝑉, {𝑣,𝑣𝑎} are linearly 𝐶-dependent. In this case standard arguments show that 𝑎∈𝐶; hence [𝑥,𝑦]^𝑚(𝛾+𝑎+𝑏)=[𝑥,𝑦]^𝑛(𝑎+𝑏) and by Lemma 6 we get 𝛾+𝑎+𝑏=𝑎+𝑏; that is, 𝛾=0 and 𝑎=−𝑏∈𝐶.

Lemma 9. Let 𝑅 be a 2-torsion free primitive ring which is isomorphic to a dense ring of linear transformations of a vector space 𝑉 over the field 𝐶 such that dim_𝐶𝑉≥3, and let 𝑎,𝑏,𝑐,𝑞∈𝑅 and 𝑚,𝑛≥1 be such that 𝑎[𝑥,𝑦]^𝑚+[𝑥,𝑦]^𝑚𝑏=𝑐[𝑥,𝑦]^𝑛+[𝑥,𝑦]^𝑛𝑞, for all 𝑥,𝑦∈𝑅. Then one of the following holds:(a)𝑎,𝑏,𝑐,𝑞∈𝐶, 𝑎+𝑏=0 and 𝑐+𝑞=0;(b)𝑎−𝑐=𝑞−𝑏∈𝐶, 𝑎+𝑏=𝑐+𝑞=0 and for all 𝑥₁,𝑥₂∈𝑅 there exists 𝛼∈𝐶 such that [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛+𝛼;(c)𝑎−𝑐=𝑞−𝑏∈𝐶 and [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛 for all 𝑥₁,𝑥₂∈𝑅.

Proof. Suppose first that dim_𝐶𝑉=𝑡 is finite. In this case 𝑅≅𝑀_𝑡(𝐶), the ring of all 𝑡×𝑡 matrices over 𝐶 with 𝑡≥3. Let 𝑒_𝑖𝑗 be the usual matrix unit and denote 𝑎=∑𝑎_𝑟𝑠𝑒_𝑟𝑠, 𝑐=∑𝑐_𝑟𝑠𝑒_𝑟𝑠, with 𝑎_𝑟𝑠,𝑐_𝑟𝑠∈𝐶. Fix 𝑖≠𝑗 and choose [𝑥,𝑦]=[𝑒_𝑖𝑗,𝑒_𝑗𝑖]=𝑒_𝑖𝑖−𝑒_𝑗𝑗. By the hypothesis 𝑎𝑒_𝑖𝑖−𝑒_𝑗𝑗^𝑚+𝑒_𝑖𝑖−𝑒_𝑗𝑗^𝑚𝑏=𝑐𝑒_𝑖𝑖−𝑒_𝑗𝑗^𝑛+𝑒_𝑖𝑖−𝑒_𝑗𝑗^𝑛𝑞 (8) and left multiplying by 𝑒_𝑘𝑘, for any 𝑘≠𝑖,𝑗, it follows that 𝑎_𝑘𝑖−𝑐_𝑘𝑖=0. This means that the matrix 𝑎−𝑐 is diagonal. Let now 𝜑 be any automorpism of 𝑅 and note that 𝜑(𝑎)[𝑥,𝑦]^𝑚+[𝑥,𝑦]^𝑚𝜑(𝑏)=𝜑(𝑐)[𝑥,𝑦]^𝑛+[𝑥,𝑦]^𝑛𝜑(𝑞), for all 𝑥,𝑦∈𝑅. Therefore 𝜑(𝑎−𝑐) must be a diagonal matrix. In particular, for any 𝑖≠𝑗, (1+𝑒_𝑖𝑗)(𝑎−𝑐)(1−𝑒_𝑖𝑗) must be a diagonal matrix. By easy computation it follows that 𝑎_𝑗𝑗−𝑐_𝑗𝑗=𝑎_𝑖𝑖−𝑐_𝑖𝑖; that is, 𝑎−𝑐=𝛼∈𝑍(𝑅). Analogously one can prove that 𝑏−𝑞=𝛽∈𝑍(𝑅).
Therefore we have that 𝛼+𝛽𝑥,𝑦^𝑚+𝑐𝑥,𝑦^𝑚+𝑥,𝑦^𝑚𝑞=𝑐𝑥,𝑦^𝑛+𝑥,𝑦^𝑛𝑞. (9)
Suppose now that dim_𝐶𝑉=∞. By Lemma 2 in [13], 𝑅 satisfies the following generalized identity 𝑎𝑥^𝑚+𝑥^𝑚𝑏−𝑐𝑥^𝑛−𝑥^𝑛𝑞. Let 𝑒∈𝐻=soc(𝑅) be any idempotent minimal element and recall that 𝐻 is generated by such idempotent elements. Since 𝐻 satisfies 𝑎𝑥^𝑚+𝑥^𝑚𝑏−𝑐𝑥^𝑛−𝑥^𝑛𝑞, then in particular 𝑎𝑒+𝑒𝑏−𝑐𝑒−𝑒𝑞=0. Thus we can easily obtain both (1−𝑒)(𝑎−𝑐)𝑒=0 and 𝑒(𝑏−𝑞)(1−𝑒)=0, which imply [𝑎−𝑐,𝑒]=0 and [𝑏−𝑞,𝑒]=0; that is, [𝑎−𝑐,𝐻]=0 and [𝑏−𝑞,𝐻]=0. Also in this case there exist 𝛼,𝛽∈𝐶 such that 𝑎−𝑐=𝛼 and 𝑏−𝑞=𝛽. Therefore in any case Identity (9) holds and the conclusion follows as an application of Lemma 8.

Lemma 10. Let 𝑅 be a 2-torsion free prime ring, 𝑎,𝑏,𝑐,𝑞∈𝑅, and let 𝑚,𝑛≥1 be such that 𝑎[𝑥,𝑦]^𝑚+[𝑥,𝑦]^𝑚𝑏=𝑐[𝑥,𝑦]^𝑛+[𝑥,𝑦]^𝑛𝑞, for all 𝑥,𝑦∈𝑅. Then either 𝑅 satisfies 𝑠₄, the standard identity of degree 4, or one of the following holds:(a)𝑎,𝑏,𝑐,𝑞∈𝐶, 𝑎+𝑏=0 and 𝑐+𝑞=0;(b)𝑎−𝑐=𝑞−𝑏∈𝐶, 𝑎+𝑏=𝑐+𝑞=0 and for all 𝑥₁,𝑥₂∈𝑅 there exists 𝛼∈𝐶 such that [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛+𝛼;(c)𝑎−𝑐=𝑞−𝑏∈𝐶 and [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛 for all 𝑥₁,𝑥₂∈𝑅.

Proof. Firstly assume that 𝑅 does not satisfy any nontrivial generalized polynomial identity. In light of [14] and by our assumption, it follows that both 𝑎𝑥,𝑦^𝑚−𝑐𝑥,𝑦^𝑛,𝑥,𝑦^𝑚𝑏−𝑥,𝑦^𝑛𝑞 (10) are trivial generalized polynomial identities of 𝑅. This means that 𝑎,𝑏,𝑐,𝑞∈𝐶, so that (𝑎+𝑏)[𝑥,𝑦]^𝑚−(𝑐+𝑞)[𝑥,𝑦]^𝑛 is a trivial generalized polynomial identity for 𝑅.
Hence 𝑎+𝑏=0 and 𝑐+𝑞=0.
Consider now the case that 𝑅 satisfies some nontrivial generalized polynomial identity. By Theorem 3 in [15] it follows that 𝑆=𝑅𝐶 is a primitive ring with soc(𝑅)≠0, where 𝐶=𝑍(𝑈) is the extended centroid of 𝑅, and the Utumi quotient ring 𝑈 is a 𝐶-algebra centrally closed. Since 𝑅 and 𝑈 satisfy the same generalized polynomial identities (see [14]), without loss of generality, we may replace 𝑅 by 𝑈 and 𝑍(𝑅) by 𝐶 and 𝑅 is a 𝐶-algebra centrally closed. Then 𝑅 is a dense ring of linear transformations of a vector space 𝑉 over 𝐶. In case dim_𝐶𝑉≤2, then 𝑅 satisfies the standard identity 𝑠₄. In case dim_𝐶𝑉≥3 we may apply Lemma 9. In any case we are done.

We consider now the more general situation.

Proof of Theorem 1. In all that follows we assume that 𝑅 does not satisfy 𝑠₄; if not we are done.
Since any generalized derivation 𝐻 of 𝑅 can be implicitly assumed to be defined on the whole 𝑈 and assumes the form 𝐻(𝑥)=𝑐𝑥+ℎ(𝑥) for some 𝑐∈𝑈 and ℎ a derivation on 𝑈, we may assume that there exist 𝑎,𝑏∈𝑈 and 𝑑,𝛿 derivations on 𝑈 such that 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥),𝐺(𝑥)=𝑏𝑥+𝛿(𝑥). (11)
Since 𝑅 and 𝑈 satisfy the same generalized polynomial identities [14] as well as the same differential identities [9], then, without loss of generality, to prove our results we may assume that 𝐹𝑥,𝑦^𝑚=𝐺𝑥,𝑦^𝑛∀𝑥,𝑦∈𝑈, (12) where 𝑑, 𝛿 are derivations on 𝑈. We divide the proof into 3 cases.
Case 1. Let 𝑑(𝑥)=[𝑐,𝑥] and 𝛿(𝑥)=[𝑞,𝑥] be both inner derivations in 𝑈, so that 𝐹(𝑥)=𝑎𝑥+[𝑐,𝑥]=(𝑎+𝑐)𝑥−𝑥𝑐 and 𝐺(𝑥)=𝑏𝑥+[𝑞,𝑥]=(𝑏+𝑞)𝑥−𝑥𝑞, for suitable elements 𝑐,𝑞∈𝑈. Thus we have (𝑎+𝑐)[𝑥,𝑦]^𝑚+[𝑥,𝑦]^𝑚(−𝑐)=(𝑏+𝑞)[𝑥,𝑦]^𝑛+[𝑥,𝑦]^𝑛(−𝑞), for all 𝑥,𝑦∈𝑈. In this case we apply Lemma 10 and we have that either 𝑅 satisfies 𝑠₄ or one of the following holds:(a)𝑎=𝑏=0 and 𝑑=𝛿=0; that is, 𝐹=𝐺=0;(b)𝑎=𝑏=0, 𝑐−𝑞∈𝐶, and 𝑑=𝛿=𝐹=𝐺; that is, 𝐹=𝐺 is the inner derivation induced by the element 𝑐; moreover for all 𝑥₁,𝑥₂∈𝑅 there exists 𝛼∈𝐶 such that [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛+𝛼;(c)[𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛, for all 𝑥₁,𝑥₂∈𝑅,𝑞−𝑐∈𝐶 (i.e., 𝑑=𝛿); moreover 𝑎=𝑏+𝛽 for a suitable 𝛽∈𝐶. In this case, our assumption implies that 𝛽[𝑥₁,𝑥₂]^𝑚=0 for all 𝑥₁,𝑥₂∈𝑅, which means 𝛽=0, since 𝑅 is not commutative. Therefore 𝑎=𝑏 and 𝐹=𝐺.
Case 2. Assume that 𝑑 and 𝛿 are 𝐶-independent modulo 𝑈-inner derivations.
By the main assumption, 𝑈 satisfies 𝑎𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑚−1}𝑥,𝑦^𝑖𝑑(𝑥),𝑦+𝑥,𝑑𝑦𝑥,𝑦^𝑗=𝑏𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝛿(𝑥),𝑦+𝑥,𝛿𝑦𝑥,𝑦^𝑗. (13)
Then by the result in [16], we have that 𝑈 satisfies 𝑎𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑚−1}𝑥,𝑦^𝑖𝑥₁,𝑦+𝑥,𝑥₂𝑥,𝑦^𝑗=𝑏𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝑥₃,𝑦+𝑥,𝑥₄𝑥,𝑦^𝑗. (14)
In particular 𝑈 satisfies the blended component _{𝑖+𝑗=𝑚−1}𝑥,𝑦^𝑖𝑥₁,𝑦,𝑥,𝑦^𝑗. (15)
Thus 𝑈 is a PI-ring and there exists 𝑀_𝑡(𝐶) satisfying (15). For 𝑥=𝑒₁₂, 𝑦=𝑒₂₁, and 𝑥₁=𝑒₃₂ we have the contradiction 𝑒₃₁=0.
Case 3. Assume finally that 𝑑 and 𝛿 are not both inner derivations (otherwise we are done by Case 1) such that 𝑑 and 𝛿 are 𝐶-dependent modulo 𝑈-inner derivations. Without loss of generality say 𝑑=𝛼𝛿+𝑎𝑑_𝑞; that is, 𝑑(𝑥)=𝛼𝛿(𝑥)+[𝑞,𝑥], for suitable 𝛼∈𝐶 and 𝑞∈𝑈. Under this assumption, we prove that a number of contradictions follow. We may assume the following:(i)𝑑,𝛿 are both nonzero derivations of 𝑅. In fact, if either 𝑑 or 𝛿 is zero then both 𝐹 and 𝐺 are inner generalized derivations of 𝑅, which is a contradiction;(ii)𝛿 is not an inner derivation, if not both 𝐹 and 𝐺 are inner generalized derivations of 𝑅, which is again a contradiction.
Suppose first that 𝛼=0; then 𝑑(𝑥)=[𝑞,𝑥]. Thus 𝑈 satisfies 𝑎𝑥,𝑦^𝑚+𝑞,𝑥,𝑦^𝑚=𝑏𝑥,𝑦^𝑛+𝛿𝑥,𝑦^𝑛; (16) that is, 𝑎+𝑞𝑥,𝑦^𝑚+𝑥,𝑦^𝑚−𝑞=𝑏𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝛿(𝑥),𝑦+𝑥,𝛿𝑦𝑥,𝑦^𝑗. (17)
By [16] it follows that 𝑈 satisfies 𝑎+𝑞𝑥,𝑦^𝑚+𝑥,𝑦^𝑚−𝑞=𝑏𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝑥₁,𝑦+𝑥,𝑥₂𝑥,𝑦^𝑗 (18) and in particular 𝑈 satisfies the blended component _{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝑥₁,𝑦𝑥,𝑦^𝑗. (19)
Therefore 𝑈 is a PI-ring, so that there exists a field 𝐾 such that 𝑈 and the matrix ring 𝑀_𝑡(𝐾) satisfy the same polynomial identities. Moreover, we may assume 𝑡≥3, since 𝑈 does not satisfy 𝑠₄. Notice that if we choose 𝑥=𝑒₁₂, 𝑦=𝑒₂₁, and 𝑥₁=𝑒₃₂ then the contradiction 𝑒₃₁=0 follows.
Then assume 𝛼≠0. Thus 𝑈 satisfies 𝑎𝑥,𝑦^𝑚+𝛼𝛿𝑥,𝑦^𝑚+𝑞,𝑥,𝑦^𝑚=𝑏𝑥,𝑦^𝑛+𝛿𝑥,𝑦^𝑛; (20) that is, 𝑎+𝑞𝑥,𝑦^𝑚+𝑥,𝑦^𝑚−𝑞+𝛼_{𝑖+𝑗=𝑚−1}𝑥,𝑦^𝑖𝛿(𝑥),𝑦+𝑥,𝛿𝑦𝑥,𝑦^𝑗=𝑏𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝛿(𝑥),𝑦+𝑥,𝛿𝑦𝑥,𝑦^𝑗. (21)
By [16] it follows that 𝑈 satisfies 𝑎+𝑞𝑥,𝑦^𝑚+𝑥,𝑦^𝑚−𝑞+𝛼_{𝑖+𝑗=𝑚−1}𝑥,𝑦^𝑖𝑥₁,𝑦+𝑥,𝑥₂𝑥,𝑦^𝑗=𝑏𝑥,𝑦^𝑛+_{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝑥₁,𝑦+𝑥,𝑥₂𝑥,𝑦^𝑗 (22) and in particular 𝑈 satisfies both 𝑎+𝑞𝑥,𝑦^𝑚+𝑥,𝑦^𝑚−𝑞=𝑏𝑥,𝑦^𝑛,𝛼_{𝑖+𝑗=𝑚−1}𝑥,𝑦^𝑖𝑥₁,𝑦+𝑥,𝑥₂𝑥,𝑦^𝑗=_{𝑖+𝑗=𝑛−1}𝑥,𝑦^𝑖𝑥₁,𝑦+𝑥,𝑥₂𝑥,𝑦^𝑗. (23)(24)
As above, 𝑈 is a PI-ring, so that there exists a field 𝐾 such that 𝑈 and the matrix ring 𝑀_𝑡(𝐾) satisfy (24). Since we may assume that 𝑡≥3, then for 𝑥=𝑒₁₂, 𝑦=𝑒₂₁, 𝑥₁=𝑒₃₂, and 𝑥₂=0 in (24) we get 𝛼𝑒₃₁=𝑒₃₁; that is, 𝛼=1. Moreover, by (23) and using Lemma 10 we have in any case 𝑞∈𝐶, that is, 𝑑(𝑥)=𝛿(𝑥), and one of the following holds. (a)𝑢^𝑚=𝑢^𝑛 for all 𝑢∈[𝑈,𝑈] and there exists 𝛽∈𝐶 such that 𝑏=𝑎+𝛽. In this case we have 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥) and 𝐺(𝑥)=(𝑎+𝛽)𝑥+𝑑(𝑥) for all 𝑥∈𝑈. By the main assumption we also get 𝑎𝑢^𝑚+𝑑(𝑢^𝑚)=𝑎𝑢^𝑚+𝛽𝑢^𝑚+𝑑(𝑢^𝑚), for all 𝑢∈[𝑈,𝑈]. Hence 𝛽𝑢^𝑚=0, which implies 𝛽=0 and 𝐹=𝐺.(b)𝑎=𝑏=0 and 𝐹=𝑑=𝛿=𝐺. This implies 𝑑(𝑢^𝑚−𝑢^𝑛)=0, for all 𝑢∈[𝑈,𝑈]. Since 𝑑≠0, by Lemma 7 we have that for all 𝑢∈[𝑈,𝑈] there exists 𝛽_𝑢∈𝐶 such that 𝑢^𝑚−𝑢^𝑛=𝛽_𝑢. Without loss of generality we consider 𝑛>𝑚 and let 𝛼 be any nonzero element of 𝐶. Denote 𝑣=[𝑥,𝑦]∈[𝑈,𝑈] and 𝑢=[𝛼^−𝑚𝑥,𝑦]=𝛼^−𝑚𝑣∈[𝑈,𝑈]. Thus there exist 𝛽_𝑢,𝛽_𝑣∈𝐶 such that 𝑣^𝑚=𝑣^𝑛+𝛽_𝑣 and 𝑢^𝑚=𝑢^𝑛+𝛽_𝑢. Hence 𝛼𝑣^𝑚=𝛼^𝑛−𝑚𝑣^𝑛+𝛽_𝑢 implying 𝛼𝑣^𝑛+𝛼𝛽_𝑣=𝛼^𝑛−𝑚𝑣^𝑛+𝛽_𝑢. (25)
In other words, for all 𝛼∈𝐶 and for all 𝑣∈[𝑈,𝑈], we have (𝛼−𝛼^𝑛−𝑚)𝑣^𝑛∈𝐶. In case 𝛼≠𝛼^𝑛−𝑚 then 𝑣^𝑛∈𝐶 for all 𝑣∈[𝑈,𝑈] and as above there exists a field 𝐾 such that 𝑈 and the matrix ring 𝑀_𝑡(𝐾) satisfy [[𝑥,𝑦]^𝑛,𝑧]=0. Since we may assume that 𝑡≥3, a contradiction follows easily. Thus 𝛼=𝛼^𝑛−𝑚 for any 𝛼∈𝐶. In this last case and by (25), it follows that 𝛼𝛽_𝑣=𝛽_𝑢, for any 𝛼≠0. Hence, for 0≠𝛼₁≠𝛼₂∈𝐶, we get 𝛼₁𝛽_𝑣=𝛼₂𝛽_𝑣; that is, 𝛽_𝑣=0 for any choice of 𝑣∈[𝑈,𝑈], which means 𝑣^𝑚=𝑣^𝑛 for all 𝑣∈[𝑈,𝑈].

Corollary 11. Let 𝑅 be a 2-torsion free prime ring and let 𝐿 be a noncentral Lie ideal of 𝑅, and let 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 be two generalized derivations associated with derivations 𝑑∶𝑅→𝑅 and 𝛿∶𝑅→𝑅, respectively. Suppose that there exist 𝑚,𝑛≥1 integers such that 𝐹(𝑢^𝑚)=𝐺(𝑢^𝑛) for all 𝑢∈𝐿. Then either 𝑅 satisfies the standard identity 𝑠₄ or one of the following holds:(a)𝐹=𝐺 is an inner ordinary derivation of 𝑅 and for all 𝑥₁,𝑥₂∈𝑅 there exists 𝛼∈𝐶 such that [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛+𝛼;(b)𝐹=𝐺 and [𝑥₁,𝑥₂]^𝑚=[𝑥₁,𝑥₂]^𝑛, for all 𝑥₁,𝑥₂∈𝑅.

Proof. Since 𝐿 is not central in 𝑅, then by [10, pages 4-5] there exists a nonzero two-sided ideal 𝐼 of 𝑅 such that [𝐼,𝐼]⊆𝐿. Therefore 𝐹([𝑥,𝑦]^𝑚)=𝐺([𝑥,𝑦]^𝑛) for all 𝑥,𝑦∈𝐼. As above we write 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥),𝐺(𝑥)=𝑏𝑥+𝛿(𝑥) (26) for suitable 𝑎,𝑏∈𝑈 and 𝑑,𝛿 derivations of 𝑅. Thus 𝐼 satisfies the differential identity 𝑎𝑥,𝑦^𝑚+𝑑𝑥,𝑦^𝑚−𝑏𝑥,𝑦^𝑛−𝛿𝑥,𝑦^𝑛. (27)
Since 𝐼, 𝑅, and 𝑈 satisfy the same differential identities (see [9]), then (27) is satisfied by 𝑈. Hence we conclude by Theorem 1.

Proof of Theorem 2. Firstly we notice that for 𝑢=𝑣 we have 𝐹(𝑢^{𝑚(𝑝+𝑞)})=𝐺(𝑢^{𝑛(𝑟+𝑠)}), for all 𝑢∈𝐿. By Corollary 11 we have that 𝑅 is a PI-ring and, if assumed that 𝑅 does not satisfy 𝑠₄, then one has 𝐹=𝐺. As remarked above, there exists an ideal 𝐼 of 𝑅 such that [𝐼,𝐼]⊆𝐿; therefore 𝐹𝑢^𝑝𝑣^𝑞𝑚−𝑣^𝑟𝑢^𝑠𝑛=0 (28) for all 𝑢,𝑣∈[𝐼,𝐼]. Since 𝑈 and 𝐼 satisfy the same differential identities, we also have that (28) is satisfied by 𝑈. By Lemma 7, and since 𝐹≠0, (𝑢^𝑝𝑣^𝑞)^𝑚−(𝑣^𝑟𝑢^𝑠)^𝑛 is central for all 𝑢,𝑣∈[𝑈,𝑈], moreover there exists a field 𝐾 such that 𝑀_𝑡(𝐾) and 𝑈 satisfy the same polynomial identities. Of course we assume that 𝑡≥3, since 𝑈 does not satisfy 𝑠₄. Fix 𝑢=[𝑒₁₂,𝑒₂₁+𝑒₃₁]=𝑒₁₁−𝑒₂₂−𝑒₃₂ and 𝑣=[𝑒₁₂,𝑒₂₁]=𝑒₁₁−𝑒₂₂.
Thus 𝑢^𝑝𝑣^𝑞𝑚=𝑒₁₁+𝛼𝑒₂₂+𝛽𝑒₃₂,𝑣^𝑟𝑢^𝑠𝑛=𝑒₁₁+𝛾𝑒₂₂, (29) where 𝛼,𝛽,𝛾∈{+1,−1}. Hence 𝑢^𝑝𝑣^𝑞𝑚−𝑣^𝑟𝑢^𝑠𝑛=𝛽𝑒₃₂∉𝑍𝑀_𝑡(𝐾), (30) which is a contradiction.

Corollary 12. Let 𝑅 be a 2-torsion free prime ring, 𝑈 its Utumi quotient ring, 𝐶 its extended centroid, 𝐿 noncentral Lie ideal of 𝑅, and 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 two generalized derivations associated with derivations 𝑑∶𝑅→𝑅 and 𝛿∶𝑅→𝑅, respectively. Suppose that there exists 𝑛≥1 integer such that 𝐹((𝑢𝑣)^𝑛)=𝐺((𝑣𝑢)^𝑛) for all 𝑢,𝑣∈𝐿. Then either 𝐹=𝐺=0 or 𝑅 satisfies the standard identity 𝑠₄ and 𝐹−𝐺 is an ordinary derivation of 𝑅.

Proof. For 𝑢=𝑣∈𝐿 in our main assumption we get 𝐹(𝑢^2𝑛)=𝐺(𝑢^2𝑛); moreover by applying Theorem 2 it follows that 𝑅 must satisfy the standard identity 𝑠₄.
Therefore 𝑈≅𝑀₂(𝐶) and we may assume that 𝑅⊆𝑀₂(𝐶), the 2×2 matrix ring over 𝐶. Since 𝐿 is not central and 𝑅 is 2-torsion free, as remarked above, it follows that [𝑅,𝑅]⊆𝐿. Hence we have that 𝐹([𝑥₁,𝑥₂]^2𝑛)=𝐺([𝑥₁,𝑥₂]^2𝑛) is satisfied by 𝑅. Let 𝑎,𝑏∈𝑈 such that 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥) and 𝐺(𝑥)=𝑏𝑥+𝛿(𝑥), for all 𝑥∈𝑅; then 𝑅 satisfies the generalized differential identity (𝑎−𝑏)𝑥₁,𝑥₂^2𝑛+(𝑑−𝛿)𝑥₁,𝑥₂^2𝑛. (31)
Firstly notice that in case 𝐹=0 then 𝑅 satisfies 𝐺([𝑥₁,𝑥₂][𝑦₁,𝑦₂]^𝑛) and by Lemma 7 it follows that 𝐺=0 unless [𝑥₁,𝑥₂][𝑦₁,𝑦₂] is central valued on 𝑅. In the latter case for [𝑥₁,𝑥₂]=𝑒₁₂ and [𝑦₁,𝑦₂]=𝑒₂₁ a contradiction follows.
Thus we assume both 𝐹≠0 and 𝐺≠0 and denote 𝐻(𝑥)=𝑑(𝑥)−𝛿(𝑥) (of course 𝐻 is a derivation of 𝑅). Here we apply again Kharchenko’s theory, using the fact that [𝑥₁,𝑥₂]^2𝑛 is central valued on 𝑅, and we prove that 𝑎=𝑏. To do this, we divide the proof into two cases.
Case  1. Let 𝐻(𝑥)=[𝑐,𝑥] be the inner derivations in 𝑅. Thus we have that (𝑎−𝑏)[𝑥,𝑦]^2𝑛=0, for all 𝑥,𝑦∈𝑅, which implies 𝑎=𝑏.
Case  2. Assume that 𝐻 is not inner. By Kharchenko’s theory in [16], and since 𝑅 satisfies (𝑎−𝑏)𝑥₁,𝑥₂^2𝑛+_{𝑖+𝑗=2𝑛−1}𝑥₁,𝑥₂^𝑖𝐻𝑥₁,𝑥₂+𝑥₁,𝐻𝑥₂𝑥₁,𝑥₂^𝑗, (32) then 𝑅 satisfies (𝑎−𝑏)𝑥₁,𝑥₂^2𝑛+_{𝑖+𝑗=2𝑛−1}𝑥₁,𝑥₂^𝑖𝑦₁,𝑥₂+𝑥₁,𝑦₂𝑥₁,𝑥₂^𝑗; (33) in particular (𝑎−𝑏)[𝑥₁,𝑥₂]^2𝑛 is a generalized identity for 𝑅, which implies again that 𝑎=𝑏.

Corollary 13. Let 𝑅 be a 2-torsion free prime ring, 𝑈 its Utumi quotient ring, 𝐶 its extended centroid, and 𝐹∶𝑅→𝑅 and 𝐺∶𝑅→𝑅 two generalized derivations associated with derivations 𝑑∶𝑅→𝑅 and 𝛿∶𝑅→𝑅, respectively. Suppose that there exists 𝑛≥1 integer such that 𝐹((𝑢𝑣)^𝑛)=𝐺((𝑣𝑢)^𝑛) for all 𝑢,𝑣∈𝑅. Then either 𝐹=𝐺=0 or 𝑅 is commutative.

Proof. By Corollary 12, if we assume that 𝐹≠0 and 𝐺≠0, it follows that 𝑅⊆𝑀₂(𝐶) and 𝐹−𝐺=𝐻 is ordinary derivation of 𝑅. In particular 𝐻(𝑥²)=0 for all 𝑥∈𝑅 and by Lemma 7 it follows that either 𝐻=0 or 𝑥²∈𝑍(𝑅), for all 𝑥∈𝑅.
In case 𝐻=0, then 𝐹=𝐺 and 𝐹((𝑢𝑣)^𝑛−(𝑣𝑢)^𝑛)=0 for all 𝑢,𝑣∈𝑅. Again by Lemma 7, since 𝐹≠0, one has (𝑢𝑣)^𝑛−(𝑣𝑢)^𝑛∈𝑍(𝑅); in particular for 𝑢=𝑒₁₁ and 𝑣=𝑒₁₁+𝑒₁₂, it follows the contradiction 𝑒₁₂∈𝑍(𝑅). Therefore 𝑅 must be commutative.
Let 𝐻≠0, so that 𝑥²∈𝑍(𝑅), for all 𝑥∈𝑅 follows by the previous argument. Thus 𝑅 is commutative and by the main assumption it follows 𝐻((𝑢𝑣)^𝑛)=0. Once again by Lemma 7, since 𝐻≠0, it follows (𝑢𝑣)^𝑛∈𝑍(𝑅), which leads to a contradiction, as remarked above.

An easy consequence of Corollary 13 is the following.

Corollary 14. Let 𝑅 be a 2-torsion free prime ring, 𝑈 its Utumi quotient ring, 𝐶 its extended centroid, and 𝐹∶𝑅→𝑅 a generalized derivation of 𝑅. Suppose that there exists 𝑛≥1 integer such that 𝐹((𝑢𝑣)^𝑛)=0 for all 𝑢,𝑣∈𝑅. Then either 𝐹=0 or 𝑅 is commutative.

3. Results in Semiprime Rings

In order to prove the main result of this section we will make use of the following facts.

Remark 15. Let 𝑅 be a semiprime ring and let 𝐹 be a generalized derivation of 𝑅 associated with derivation 𝑑. If 𝐹(𝑅)=0, then 𝑑(𝑅)=0.

Claim 1 (see [17, Proposition 2.5.1]). Any derivation of a semiprime ring 𝑅 can be uniquely extended to a derivation of its left Utumi quotient ring 𝑈, and so any derivation of 𝑅 can be defined on the whole 𝑈.

Claim 2 (see [18, page 38]). If 𝑅 is semiprime then so is its left Utumi quotient ring.
The extended centroid 𝐶 of a semiprime ring coincides with the center of its left Utumi quotient ring.

Claim 3 (see [18, p. 42]). Let 𝐵 be the set of all the idempotents in 𝐶, the extended centroid of 𝑅. Assume that 𝑅 is B-algebra orthogonal complete. For any maximal ideal 𝑃 of 𝐵, 𝑃𝑅 forms a maximal prime ideal of 𝑅, which is invariant under any derivation of 𝑅.

We are now ready to prove Theorems 3 and 4.

Proof of Theorem 3. Let 𝑎,𝑏∈𝑈 and 𝑑,𝛿 derivations of 𝑈 be such that 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥) and 𝐺(𝑥)=𝑏𝑥+𝛿(𝑥), for all 𝑥∈𝑅. By Claim 2, 𝑍(𝑈)=𝐶, and by Claim 1  𝑑 and 𝛿 can be uniquely defined on the whole 𝑈. Since 𝑈 and 𝑅 satisfy the same generalized differential identities, then 𝐹((𝑥𝑦)^𝑛)=𝐺((𝑦𝑥)^𝑛), for all 𝑥,𝑦∈[𝑈,𝑈]. Let 𝐵 be the complete Boolean algebra of idempotents in 𝐶 and 𝑀 any maximal ideal of 𝐵.
Since 𝑈 is 𝐵-algebra orthogonal complete (see [18, page 42, (2) of Fact 1]), by Claim 3, 𝑀𝑈 is a prime ideal of 𝑈, which is both 𝑑-invariant and 𝛿-invariant. Let 𝑑 and 𝛿 be the derivations induced, respectively, by 𝑑 and 𝛿 on 𝑈=𝑈/𝑀𝑈 and denote 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥), 𝐺(𝑥)=𝑏𝑥+𝛿(𝑥). For any 𝑥,𝑦∈[𝑈,𝑈], 𝐹((𝑥𝑦)^𝑛)=𝐺((𝑦𝑥)^𝑛). In particular 𝑈 is a prime ring and so, by Corollary 12, either 𝐹=0 and 𝐺=0 in 𝑈 or 𝑈 satisfies 𝑠₄. This implies that, for any maximal ideal 𝑀 of 𝐵, 𝐹(𝑈)⊆𝑀𝑈 and 𝐺(𝑈)⊆𝑀𝑈 or 𝑠₄(𝑥₁,𝑥₂,𝑥₃,𝑥₄)⊆𝑀𝑈, for all 𝑥₁,𝑥₂,𝑥₃,𝑥₄∈𝑈. In any case 𝐹(𝑈)𝑠₄(𝑥₁,𝑥₂,𝑥₃,𝑥₄)⊆⋂_𝑀𝑀𝑈=0 and also 𝐺(𝑈)𝑠₄(𝑥₁,𝑥₂,𝑥₃,𝑥₄)⊆⋂_𝑀𝑀𝑈=0. From [17, Chapter 3] there exists a central idempotent element 𝑒 of 𝑈, such that 𝐹(𝑒𝑈)=0, 𝐺(𝑒𝑈)=0 and (1−𝑒)𝑈 satisfies 𝑠₄.

Proof of Theorem 4. Here we repeat the same argument above. Let 𝑎,𝑏∈𝑈 such that 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥) and 𝐺(𝑥)=𝑏𝑥+𝛿(𝑥). Let 𝐵 be the complete Boolean algebra of idempotents in 𝐶 and 𝑀 any maximal ideal of 𝐵.
Since 𝑈 is 𝐵-algebra orthogonal complete (see [18, p. 42, (2) of Fact 1]), by Claim 3, 𝑀𝑈 is a prime ideal of 𝑈, which is both 𝑑-invariant and 𝛿-invariant. Let 𝑑 and 𝛿 be two derivations induced, respectively, by 𝑑 and 𝛿 on 𝑈=𝑈/𝑀𝑈 and denote 𝐹(𝑥)=𝑎𝑥+𝑑(𝑥), 𝐺(𝑥)=𝑏𝑥+𝛿(𝑥). For any 𝑥,𝑦∈[𝑈,𝑈], 𝐹((𝑥𝑦)^𝑛)=𝐺((𝑦𝑥)^𝑛). Since 𝑈 is a prime ring so, by Corollary 13, either 𝐹=0 and 𝐺=0 in 𝑈 or 𝑈 is commutative. Moreover, if 𝐹=0 and 𝐺=0 in 𝑈 then, by Remark 15, it follows that 𝑑=0 and 𝛿=0 in 𝑈.
This implies that, for any maximal ideal 𝑀 of 𝐵, one of the following holds:(a)either 𝐹(𝑈)⊆𝑀𝑈, 𝑑(𝑈)⊆𝑀𝑈, 𝐺(𝑈)⊆𝑀𝑈 and 𝛿(𝑈)⊆𝑀𝑈;(b)or [𝑈,𝑈]⊆𝑀𝑈.
In any case [𝐹(𝑈),𝑈]⊆⋂𝑀𝑈=0 and [𝑑(𝑈),𝑈]⊆⋂𝑀𝑈=0 imply that [𝑎𝑈,𝑈]=0. Analogously [𝐺(𝑈),𝑈]⊆⋂𝑀𝑈=0 and [𝛿(𝑈),𝑈]⊆⋂𝑀𝑈=0 implying that [𝑏𝑈,𝑈]=0.
As mentioned above it follows that there exists a central idempotent element 𝑒 in 𝑈 such that, on the direct sum decomposition 𝑒𝑈⊕(1−𝑒)𝑈, 𝐹 and 𝐺 vanish identically on 𝑒𝑈 and the ring (1−𝑒)𝑈 is commutative.
Moreover, [𝑎𝑈²,𝑈]=(0) implies 𝑎𝑈[𝑈,𝑈]=(0); that is, both 𝑈𝑎𝑈²[𝑈,𝑈]=(0) and 𝑎𝑈𝑈²[𝑈,𝑈]=(0). Therefore [𝑎,𝑈]𝑈²[𝑎,𝑈]=(0) and, by the semiprimeness of 𝑈, it follows [𝑎,𝑈]=(0); that is, 𝑎∈𝐶. Analogously one can prove that 𝑏∈𝐶.
On the other hand, since by Theorem 3 in [9] 𝑈 and 𝑅 satisfy the same differential identities, then [𝑑(𝑅),𝑅]=0 and [𝛿(𝑅),𝑅]=0, which imply that 𝑑(𝑅)⊆𝑍(𝑅), 𝛿(𝑅)⊆𝑍(𝑅), and 𝑅 contains some nonzero central ideals, unless when 𝑑(𝑅)=0 and 𝛿(𝑅)=0.
In the last case, 𝜆(𝑢𝑣)^𝑛=𝜇(𝑣𝑢)^𝑛 for all 𝑢,𝑣∈𝑅, and in particular 𝜆(𝑢)^2𝑛=𝜇(𝑢)^2𝑛 for all 𝑢∈𝑅. The semiprimeness of 𝑅 forces 𝜆=𝜇≠0 (since 𝐹≠0), so that (𝑢𝑣)^𝑛=(𝑣𝑢)^𝑛 for all 𝑢,𝑣∈𝑅. Finally by Theorem 1 in [19], the commutativity of 𝑅 follows which is a contradiction.

Conflict of Interests

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