On generalized $(\alpha, \ast)$-derivations and $\alpha$-centralizers  on Rings

The intention of the current research is to define the concept of $(\alpha,\ast)$-derivations on ring $\mathcal{R}$, where $\alpha$ is an automorphism of $\mathcal{R}$ and $\ast$ represents involution on $\mathcal{R}$. We obtain some commutativity theorems in case of prime ring by utilizing the role of $\alpha$ and $\ast$. We will also discuss the proofs of theorems in case of non-commutative prime ring and under which condition generalized $(\alpha,\ast)$-derivation behaves like an $\alpha$-centralizers. Suitable examples are given in favor of introduced concept.