On semiprime rings with multiplicative (generalized)-derivations

Let (Formula presented.) be a semiprime ring. A mapping F : R (Formula presented.) R (not necessarily additive) is said to be a multiplicative (generalized)-derivation if (Formula presented.) holds for all (Formula presented.) , where (Formula presented.) is any map (not necessarily a derivation nor an additive). The objective of the present paper is to study the following situations (i) (Formula presented.) ; (ii) (Formula presented.) ; (iii) (Formula presented.) ; (iv) (Formula presented.) ; (v) (Formula presented.) ; (vi) (Formula presented.) , for all (Formula presented.) in some appropriate subsets of (Formula presented.). © 2015, The Managing Editors.