On Ideals and Multiplicative (Generalized) - $(\Phi,\Phi)$ - Derivations

Abstract

Let $P$ be a prime ring. $I$ is a nonzero ideal of $P$. $\Phi$ is an automorphism on P. A mapping $M:P \to P$ is called Multiplicative (generalized) $(\Phi, \Phi)$-derivation if there exist a map $d: P\to P$ such that $M(a, b)=M(a) \Phi(b)+\Phi(a) d(b)$ holds for all $a, b \in P$. The objective of the present paper is to study the following identities (i). If $M(ab)+M(a) M(b)=0$ for all $a, b \in I$ then $\Phi(I)[M(a), M(b)]=0$ for all $a \in I$ (ii). Let $M_{1}$ and $M_{2}$ be two multiplicative (generalized)-$(\Phi, \Phi)$ derivations on P associated with the maps $d_{1}$ and $d_{2}$ on P respectively. If $M_{1}(a b)=\Phi(b) \circ M_{2}(a)$ for all $a, b \in I$ then $R$ is abelian or commutative or $\Phi(I)\left[\Phi(I), M_{2}(I)\right]=0$ (iii). If $M_{1}(ab)=\left[\Phi(b), M_{2}(a)\right]$ for all $a, b \in I$ then either $\Phi(I)\left[\Phi(I), M_{2}(I)\right]=(0)$ or $R$ is commutative.