Generalized derivations

In this work, we compile a rigorous investigation of generalized derivations 𝐹: 𝑅 → 𝑅 acting on nonzero left ideals 𝐿 of prime rings 𝑅. These generalized derivations, associated with derivations 𝐷, extend ordinary derivation frameworks and satisfy 𝐹(𝑥𝑦) = 𝐹(𝑥)𝑦 + 𝑥𝐷(𝑦), for all 𝑥, 𝑦 ∈ 𝐿,providing an advanced algebraic tool to examine the structural properties of rings. The study focuses on conditions such as 𝐹(𝑥)𝑦 + 𝑥𝐷(𝑦) = 0, for all 𝑥, 𝑦 ∈ 𝐿,which impose annihilation constraints on elements of left ideals, establishing essential links between generalized derivations and commutativity in prime rings. Through detailed analysis and extensions of foundational results, including criteria inspired by Posner’s theorem, we identify conditions under which generalized derivations must either vanish, enforce commutativity of 𝑅, or require 𝐿 to be contained in the center 𝑍(𝑅). Furthermore, the project explores cryptographic applications by constructing nonlinear S-boxes using generalized derivations on upper-triangular matrix rings 𝑇n(𝐹q). These constructions produce nonlinear, diffusive mappings with high algebraic complexity, demonstrating the practical potential of generalized derivations in secure cryptographic design. Overall, this work consolidates theoretical insights on left ideals of prime rings with practical constructions, highlighting the structural and applicative significance of generalized derivations.

Let τ and σ : X-→ X be automorphisms of an arbitrary associative ring X, and let L be a prime ideal of X. The main objective of this article is to combine the notions of generalized L-derivations and (τ, σ)-L-derivations by introducing and analyzing a novel additive mapping Π : X → X called a generalized (τ, σ)-L-derivation associated with a (τ, σ)-L-derivation π. Later, we will examine the algebraic properties of a factor ring X/L under the influence of certain algebraic expressions containing this generalized (τ, σ)-Lderivation and lying in a prime ideal L. Through our main findings, we establish certain results under different conditions. It also provides various illustrative examples to show that our primeness hypotheses in various theorems are not exaggerated.

In current article, for a prime ideal P of any ring R, we study the commutativity of the factor ring R/P, whenever R equipped with generalized reverse derivations F and G associated with reverse derivations d and g, respectively. That satisfies certain differential identities involving in P that connected to an ideal of R. Additionally, we show that, for some cases, the range of the generalized reverse
derivation F or G repose in the prime ideal P. Moreover, we explore several consequences and special cases. Throughout, we provide examples to demonstrate that various restrictions in the assumptions of our
outcomes are essential.

The main goal of this article is to delve deeper into the discussion of the commutativity of a factor ring R/P by analyzing certain differential identities that involve generalized P-derivations and P-multipliers connecting I to P. Here, I represents a non-zero ideal of an arbitrary ring R, and P is a prime ideal of R such that P ⊊ I. Furthermore, we will explore some outcomes from our various theorems. To underscore the necessity of the primeness assumption in our theorems, we will provide some illustrative counterexamples.

The purpose of this paper is to examine the behavior of a factor ring R/P with a partial range I of a ring R, where P is a prime ideal of R and I is a non-zero ideal of R such that P ⊊ I. In order to accomplish this objective, we will utilize specific algebraic identities that involve P and are related to generalized (α, β)-derivations F and G, associated with (α, β)-derivations d and g, respectively, where α and β are automorphisms of R. Additionally, we will list some important related consequences. Finally, we will provide an illustrative example to emphasize the importance of the hypotheses imposed in our theorems.

The current article focuses on studying the behavior of a ring ℜ/Π when ℜ
admits generalized derivations Ψ and Ω with associated derivations ϕ and δ, respectively.
These derivations satisfy specific differential identities involving Π, where Π is a prime
ideal of an arbitrary ring ℜ, not necessarily prime or semiprime. Furthermore, we explore
some consequences of our findings. To emphasize the necessity of the primeness of Π in
the hypotheses of our various theorems, we provide a list of examples.

T his paper gives the notion of orthogonality between the derivation and biderivation of a semiprime nonassociative accessible ring. We prove that if R is a 2-divisible semiprime accessible ring, B is a biderivation and D is a derivation of R, then B and D are orthogonal if and only if any one of the following equivalent conditions holds for every x, y I R: .......

Let R be a prime ring of characteristic dierent from 2, L a non-central Lie ideal of R, and m; n xed positive integers. If R admits a generalized derivation F associated with a deviation d such that 􀀀 F(u)2m 􀀀 (F(u))2n 2 Z(R) for all u 2 L, then R satises s4, the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

In this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential.

In this article we study certain differential identities in prime and semiprime rings. In particular, we prove if a prime rings satisfies certain differential identity on a nonzero ideal of a ring R, then R is commutative. Finally, as an application we obtain some range inclusion results of continuous generalized derivations on non-commutative Banach algebras.

Let R be a non-commutative prime ring and I a non-zero left ideal of R. Let U be the left Utumi quotient ring of R and C be the center of U and k, m, n, r fixed positive integers. If there exists a generalized derivation g of R such that [g(x m)x n , x r ] k = 0 for all x ∈ I, then there exists a ∈ U such that g(x) = xa for all x ∈ R except when R ∼ = M 2 (GF (2)) and I[I, I] = 0.

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

The concept of derivation in incline algebra was introduced by N.O.Alsherhi[1]. Kyung Ho kim and so Young Park[2] introduced the symmetric bi-f-derivation in incline algebra. In this paper, we introduce the concept of symmetric bit derivation in incline algebras and present some properties of symmetric bi-tderivations. Also, we characterize () and () by symmetric bit derivations in incline algebra and give some examples. Also we define isotone symmetric bit derivation in incline algebra and analyse its properties.

Let R be a ring and U 6= 0 be a Lie ideal of R. A bi-additive symmetric map B(., .) : R×R→ R is called symmetric bi-derivation if, for any y ∈ R, the map x 7→ B(x, y) is a derivation. A mapping f : R → R defined by f(x) = B(x, x) is called the trace of B. In the present paper, we shall show that U ⊆ Z(R) such that R is a prime and semiprime ring admitting the trace f satisfying the several conditions of symmetric bi-derivation.

The research led us to a generalized formulation for the N-Pendulum [Simple] system, encompassing both dragless and drag-inclusive scenarios. The mathematical framework presented herein provides a comprehensive and rigorous basis for understanding the dynamic behavior of such pendulum configurations.

Our curiosity led us to explore the behavior of an infinite pendulum system. Subsequently, we engaged in a research.

Let R be a prime ring of characteristics different from 2 and 3. If there exits a nonzero derivation d from R to itself that the map x → [[[[d(x), x], x], x], x] is centralizing on R then d = 0. Combining this result together withthe result of Sinclair and Johnson, we extend the Singer-Wermer theorem and its application in Banach algebra. Finally,we conclude some open problems. Mathematics Subject Classifications: 16A12, 16A70, 16W25, 16N60, 46K15

Let be a semiprime ring with characteristic not two, a nonzero ideal of , and , are two an epimorphism of. An additive mapping : → is generalized (,)-derivation on if there exists a (,)-derivation : → such that () = () () + () (), holds for all , ∈ and () () ≠ 0. In this paper, if satisfies any one of the following conditions:(i) ([ , ]) = ±() , ; (ii) () = ±[ , ] , ; (iii) [ (), ] , = (()) , ; (iv) [ (), ()] , = 0; (v) [ (), ()] , = (); (vi) () () = ±[ , ] , ; (vii) () − () ∈ , ; (viii) () + () ∈ , , for all , ∈. Then we prove that contains a nonzero central ideal, that is ⊆ ().

The present paper deals with the commutativity of an associative ring R and a unital Banach Algebra A via derivations. Precisely, the study of multiplicative (generalized)-derivations F and G of semiprime (prime) ring R satisfying the identities G(xy) ± [F (x), y] ± [x, y] ∈ Z(R) and G(xy) ± [x, F (y)] ± [x, y] ∈ Z(R) has been carried out. Moreover, we prove that a unital prime Banach algebra A admitting continuous linear generalized derivations F and G is commutative if for any integer n > 1 either G((xy) n) + [F (x n), y n ] + [x n , y n ] ∈ Z(A) or G((xy) n) − [F (x n), y n ] − [x n , y n ] ∈ Z(A).

Let be a semiprime ring with characteristic not two, a nonzero ideal of , and , are two an epimorphism of. An additive mapping : → is generalized (,)-derivation on if there exists a (,)-derivation : → such that () = () () + () (), holds for all , ∈ and () () ≠ 0. In this paper, if satisfies any one of the following conditions:(i) ([ , ]) = ±() , ; (ii) () = ±[ , ] , ; (iii) [ (), ] , = (()) , ; (iv) [ (), ()] , = 0; (v) [ (), ()] , = (); (vi) () () = ±[ , ] , ; (vii) () − () ∈ , ; (viii) () + () ∈ , , for all , ∈. Then we prove that contains a nonzero central ideal, that is ⊆ ().

Let R be a 2-torsion free prime ring with center Z(R) , J be a nonzero Jordan ideal also a subring of R , and F be a generalized derivation with associated derivation d. In the present paper, we shall show that J ⊆ Z(R) if any one of the following properties holds: (i) [F (u), u] ∈ Z(R) , (ii) F (u)u = ud(u) , (iii) d(u 2) = 2F (u)u , (iv) F (u 2) − 2uF (u) = d(u 2) − 2ud(u) , (v) F 2 (u) + 3d 2 (u) = 2F d(u) + 2dF (u) , (vi) F (u 2) = 2uF (u) for all u ∈ J .

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

In this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential.

Let R be an associative ring with center Z(R) and a nonzero ideal I. Let G and F are multiplicative (generalized)derivations of R together with mappings g and f respectively. In this note, we prove that R contains a non-zero central ideal if any one of the following holds for all x, y in I: 1. G(xy) ± F(x)F(y) ± [x,y] ϵ Z(R) 2. G(xy) ± F(x)F(y) ± (xoy) ϵ Z(R) 3. G(xy) ± F(x)F(y) ± yx ϵ Z(R) 4. G(xy) ± F(x)F(y) ± xy ϵ Z(R) Moreover, we investigate the identities F(x)F(y) ± yx ϵ Z(R) and F(xy) ± yx ϵ Z(R) over a non-zero left ideal of R and improve some known results.

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

Let R be a prime ring with extended centroid C, F a generalized derivation of R and n ≥ 1, m ≥ 1 fixed integers. In this paper we study the situations: 1. (F (x • y)) m = (x • y) n for all x, y ∈ I, where I is a nonzero ideal of R; 2. (F (x • y)) n = (x • y) n for all x, y ∈ I, where I is a nonzero right ideal of R. Moreover, we also investigate the situation in semiprime rings and Banach algebras.

Let 1 < k and m,k ∈ ℤ+. In this manuscript, we analyse the action of (semi)-prime rings satisfying certain differential identities on some suitable subset of rings. To be more specific, we discuss the behaviour of the semiprime ring ℛ satisfying the differential identities ([d([s,t]m), [s,t]m])k = [d([s,t]m), [s,t]m] for every s,t ∈ℛ.

Let R be a 2-torsion free prime ring with center Z, right Utumi quotient ring U , generalized derivation F associated with a nonzero derivation d of R and L a Lie ideal of R. If F (uv) n = F (u) m F (v) l or F (uv) n = F (v) l F (u) m for all u, v ∈ L, where m, n, l are fixed positive integers, then L ⊆ Z. We also examine the case where R is Ahmad N. Alkenani et al. a semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.

Let R be a prime ring of characteristic not 2 and let I be a nonzero right ideal of R. Let U be the right Utumi quotient ring of R and let C be the center of U. If G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ I, then R is commutative or there exist a, b ∈ U such that G(x) = ax + xb for all x ∈ R and one of the following assertions is true: (1) (a − λ)I = (0) = (b + λ)I for some λ ∈ C, (2) (a − λ)I = (0) for some λ ∈ C and b ∈ C. Нехай R-просте кiльце, характеристика якого не дорiвнює 2, а I-ненульовий правий iдеал R. Нехай Uправе фактор-кiльце Утумi кiльця R, а C-центр U. Якщо G є узагальненим диференцiюванням R таким, що [[G(x), x], G(x)] = 0 для всiх x ∈ I, то R є комутативним або iснують a, b ∈ U такi, що G(x) = ax + xb для всiх x ∈ R i виконується одне з наступних тверджень: (1) (a − λ)I = (0) = (b + λ)I для деякого λ ∈ C, (2) (a − λ)I = (0) для деяких λ ∈ C та b ∈ C. 1. Introduction. Throughout this paper R will always denote a prime ring with center Z(R), extended centroid C, right Utumi quotient ring U (sometimes, as in [2], U is called the maximal right ring of quotients), and two-sided Martindale quotient ring Q (see [2] for the definitions). For any x, y ∈ R, the commutator of x and y is denoted by [x, y] and defined to be xy − yx. An additive mapping d from R into itself is called a derivation of R if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. An additive mapping g : R → R is called a generalized derivation of R if there exists a derivation d of R such that g(xy) = g(x)y + xd(y) for all x, y ∈ R [10]. Obviously any derivation is a generalized derivation. Moreover, other basic examples of generalized derivations are the mappings of the form x → ax + xb, for a, b ∈ R. A generalized derivation in this form is called (generalized) inner. Many authors have studied generalized derivations in the context of prime and semiprime rings (see [1, 10, 13, 14]). In [13], T. K. Lee extended the definition of a generalized derivation as follows. By a generalized derivation he means an additive mapping g : I → U such that g(xy) = g(x)y +xd(y) for all x, y ∈ I, where I is a dense right ideal of the prime ring R and d is a derivation from I into U. He also proved that every generalized derivation can be uniquely extended to a generalized derivation of U, and moreover, there exist a ∈ U and a derivation d of U such that g(x) = ax + d(x) for all x ∈ U [13] (Theorem 3). In [7], De Filippis proved that if R is a prime ring of characteristic not 2 and G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ R, then either R is commutative or there exists λ ∈ C such that G(x) = λx for all x ∈ R. In the same paper, he uses his result to prove a theorem concerning noncommutative Banach algebras. More precisely, he proves the following:

In this article we study certain differential identities in prime and semiprime rings. In particular, we prove if a prime rings satisfies certain differential identity on a nonzero ideal of a ring R, then R is commutative. Finally, as an application we obtain some range inclusion results of continuous generalized derivations on non-commutative Banach algebras.

Let R be an associative ring. An additive mapping d : R ! R is called a Jordan derivation if dðx 2 Þ ¼ dðxÞx þ xdðxÞ holds for all x 2 R. The objective of the present paper is to characterize a prime ring R which admits Jordan derivations d and g such that ½dðx m Þ; gðy n Þ ¼ 0 for all x; y 2 R or dðx m Þ gðy n Þ ¼ 0 for all x; y 2 R, where m P 1 and n P 1 are some fixed integers. This partially extended Herstein's result in [6, Theorem 2], to the case of (semi)prime ring involving pair of Jordan derivations. Finally, we apply these purely algebraic results to obtain a range inclusion result of continuous linear Jordan derivations on Banach algebras.

In this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential.

Let R be a prime ring of characteristic dierent from 2, L a non-central Lie ideal of R, and m; n xed positive integers. If R admits a generalized derivation F associated with a deviation d such that 􀀀 F(u)2m 􀀀 (F(u))2n 2 Z(R) for all u 2 L, then R satises s4, the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

In this paper, we introduce the notion of generalized left derivation on a ring R and prove that every generalized Jordan left derivation on a 2-torsion free prime ring is a generalized left derivation on R. Some related results are also obtained.

LetRbe a ring andSa nonempty subset ofR. Suppose thatθandϕare endomorphisms ofR. An additive mappingδ:R→Ris called a left(θ,ϕ)-derivation (resp., Jordan left(θ,ϕ)-derivation) onSifδ(xy)=θ(x)δ(y)+ϕ(y)δ(x)(resp.,δ(x2)=θ(x)δ(x)+ϕ(x)δ(x)) holds for allx,y∈S. Suppose thatJis a Jordan ideal and a subring of a2-torsion-free prime ringR. In the present paper, it is shown that ifθis an automorphism ofRsuch thatδ(x2)=2θ(x)δ(x)holds for allx∈J, then eitherJ⫅Z(R)orδ(J)=(0). Further, a study of left(θ,θ)-derivations of a prime ringRhas been made which acts either as a homomorphism or as an antihomomorphism of the ringR.

In this paper, we introduce the notion of generalized left derivation on a ring R and prove that every generalized Jordan left derivation on a 2-torsion free prime ring is a generalized left derivation on R. Some related results are also obtained.

Let R be a 2-torsion free ring and U be a square closed Lie ideal of R. Suppose that α,β are automorphisms of R. An additive mapping δ:R→R is said to be a Jordan left (α,β)-derivation of R if δ(x 2 )=α(x)δ(x)+β(x)δ(x) holds for all x∈R. In this paper it is established that if R admits an additive mapping G:R→R satisfying G(u 2 )=α(u)G(u)+α(u)δ(u) for all u∈U and a Jordan left (α,α)-derivation δ; and U has a commutator which is not a left zero divisor, then G(uv)=α(u)G(v)+α(v)δ(u) for all u,v∈U. Finally, in the case of prime ring R it is proved that if G:R→R is an additive mapping satisfying G(xy)=α(x)G(y)+β(y)δ(x) for all x,y∈R and a left (α,β)-derivation δ of R such that G also acts as a homomorphism or as an anti-homomorphism on a nonzero ideal I of R, then either R is commutative or δ=0 on R.

Let R be a ring and S be a nonempty subset of R. A mapping f : R −→ R is said to be centralizing (resp. commuting) on S if [x, f (x)] ∈ Z(R) (resp. [x, f (x)] = 0) for all x ∈ S. The purpose of this paper is to generalize the classical theorem of Posner [7, Theorem 2] and to extend a result of Bell and Martindale [1, Theorem 3] for a generalized derivation of a semiprime ring R which is commuting on a left ideal of R.

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mapping F : R −→ R is called a generalized (θ, φ)-derivation on S if there exists a (θ, φ)derivation d : R −→ R such that F (xy) = F (x)θ(y) + φ(x)d(y), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U. The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R).

Let R be a noncommutative prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f (x1,. .. , xn) be a non-central multilinear polynomial over C, m ≥ 1 a fixed integer, a a fixed element of R, G a non-zero generalized derivation of R. If aG(f (r1,. .. , rn)) m ∈ Z(R) for all r1,. .. , rn ∈ I, then one of the following holds: (1) aI = aG(I) = (0); (2) G(x) = qx, for some q ∈ U and aqI = 0; (3) [f (x1,. .. , xn), xn+1]xn+2 is an identity for I; (4) G(x) = cx + [q, x] for all x ∈ R, where c, q ∈ U such that cI = 0 and [q, I]I = 0; (5) dimC(RC) ≤ 4; (6) G(x) = αx, for some α ∈ C; moreover a ∈ C and f (x1,. .. , xn) m is central valued on R.