On centrally-extended multiplicative (generalized)-(α, β)-derivations in semiprime rings

Let R be a ring with center Z and alpha, beta and d mappings of R. A mapping F of R is called a centrally-extended multiplicative (generalized)-(alpha, beta)-derivation associated with d if F(xy) - F(x)alpha(y) - beta(x)d(y) is an element of Z for all x, y is an element of R. The objective of the present paper is to study the following conditions: (i) F(xy) +/- beta(x)G(y) is an element of Z, (ii) F(xy) +/- g(x)alpha(y) is an element of Z and (iii) F(xy) +/- g(y)alpha(x) is an element of Z for all x, y in some appropriate subsets of R, where G is a multiplicative (generalized)-(alpha, beta)-derivation of R associated with the map g on R.