Structure of a quotient ring R/P with generalized derivations acting on the prime ideal P and some applications

The main purpose of this paper is to develop a new approach that consists to investigate the structure of a quotient ring R / P via action of generalized derivations on the prime ideal P . In this direction, we initiate new classes of additive mappings extending both of centralizing and commuting mappings. Furthermore, for an arbitrary ring, we will consider algebraic identities based on its prime ideals. As an application, we obtain some results concerning invariance of minimal prime ideals of a semiprime ring under derivations.