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Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.

Let ℧ be a prime ring of char(℧) ≠ 2 with its center Z. This article introduces new classes of endomorphisms and investigates how they relate to antiautomorphisms of prime rings and the commutativity of prime rings. Additionally, we fully describe and classify some of these endomorphisms. We also give examples to prove the necessity of the numerous restrictions included in the hypotheses of our results. [ABSTRACT FROM AUTHOR]

This article examines the commutativity of rings with antiautomorphisms, specifically when they are equipped with derivations that satisfy certain algebraic identities. Moreover, we present examples to demonstrate the necessity of the various restrictions imposed in the hypotheses of our theorems. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, L a non-central Lie ideal of R (i.e., L , R ≠ 0 ), n ≥ 1 a fixed integer, F and G two generalized skew derivations of R, associated with the same automorphism α of R. If (F (x y) − G (x) y) n = 0 , for any x , y ∈ L , then there exists a ∈ Q r such that , for any x ∈ R , unless when R satisfies the standard polynomial identity s 4 (x 1 , ... , x 4) . [ABSTRACT FROM AUTHOR]

Let A be a non-commutative prime ring with involution ∗ , of characteristic ≠ 2 (and 3) , with Z as the center of A and Π a mapping Π : A → A such that [ Π (x) , x ] ∈ Z for all (skew) symmetric elements x ∈ A. If Π is a non-zero CE-Jordan derivation of A , then A satisfies s 4 , the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A , then A satisfies s 4 or Π (y) = λ (y − y *) for all y ∈ A , and some λ ∈ C , the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the commutativity of a prime ring ℛ with anti-automorphism ψ and generalized derivation (ℱ , ξ) satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential. [ABSTRACT FROM AUTHOR]

The present paper's primary purpose is to study the connection between commutativity of rings and the behavior of its derivations in prime rings. [ABSTRACT FROM AUTHOR]

Let X be a real or complex Banach space, let L ( X ) be the algebra of all bounded linear operators of X and let A ( X ) ⊆ L ( X ) be a standard operator algebra. Suppose there exists a linear mapping T : A ( X ) → L ( X ) satisfying the relation T ( A n ) = T ( A ) A n −1 − AT ( A n −2 ) A − A n −1 T ( A ) for all A ∈ A ( X ) , where n > 2 is some fixed integer. Then T is of the form: (i) T ( A ) = 0 for all A ∈ F ( X ) and (ii) T ( A ) = BA , for all A ∈ A ( X ) and some B ∈ L ( X ) . [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f (x1, . . . , xn) a multilinear polynomial over C, ρ a nonzero right ideal of R andm > 1 a fixed integer such that ([d(f(r1, . . . , rn)), f(r1, . . . , rn)])m = [d(f(r1, . . . , rn)), f(r1, . . . , rn)] for all r1, . . . , rn ∈ ρ. Then either [f(x1,..., xn), xn+1]xn+2 is an identity for ρ or d(ρ)ρ = 0. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic not 2, U a nonzero square closed Lie ideal of R and F;G;H be the generalized derivations with associated derivations d,δh of R respectively. In the present paper, we study the situations if one the follwoing holds (1) F(u)G(v)±H(uv)ϵ Z(R), (2) F(u)F (v)±H(vu) ϵ Z(R), for all u,v ϵ U, then U ⊆Z(R). [ABSTRACT FROM AUTHOR]

Let R be a prime ring, I a nonzero ideal of R, and a ε R. Suppose that σ is a nontrivial automorphism of R such that af{(σ(x 0 y))n - (x 0 y)m} = 0 or a{(σ([x; y]))n - ([x; y])m} = 0 for all x,y ε I, where n and m are fixed positive integers. We prove that if char(R) > n + 1 or char(R) = 0, then either a = 0 or R is commutative. [ABSTRACT FROM AUTHOR]

Let R be a prime ring and set [x,y]1 = [x,y] = D xy - yx for all x,y ∊ R and inductively [x,y]k = [[x,y]k-1,y] for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that (δ,φ) is a skew derivation of R such that δ([x,y])= [x,] = n for all x;y ∊ I , then R is commutative. [ABSTRACT FROM AUTHOR]