A NOTE ON MULTIPLICATIVE (GENERALIZED) (α, β)-DERIVATIONS IN PRIME RINGS

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