Annihilator on prime rings admitting multiplicative generalized g-derivations

Abstract

Suppose $\Re$ is a ring and $g:\Re \rightarrow Q_{r}$ be an arbitrary map. An additive map $d:\Re \rightarrow Q_{r}$ is said to be g-derivation if $d(xy) = d(x)y+g(x)d(y)$  holds $~ \text{ for } \text{ all }~ x,y\in \Re .$ An additive map $G:\Re \rightarrow Q_{r}$ is said to be generalized g-derivation if $G(xy) = G(x)y+g(x)d(y)$  holds $~ \text{ for } \text{ all }~ x,y\in \Re .$ For any subset S of $\Re$, $S\subseteq \Re$. The left annihilator of S in $\Re$ is denoted by $l_{\Re }(S)$ and defined by $l_{\Re }(S) = \{x\in \Re \mid xS = 0\}.$ In the present paper, we study the left annihilator identities on prime rings admitting multiplicative generalized g-derivations.