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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 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]

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]

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]