Annihilator on prime rings admitting multiplicative generalized g-derivations

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.