Identities with Multiplicative (Generalized) $(\alpha,\beta)$-derivations in Semiprime Rings

Abstract

Let $R$ be an associative ring and $\alpha,\beta$ be automorphisms on $R$. A mapping $F:R\rightarrow R$ (not necessarily additive) is said to be a multiplicative (generalized)-$(\alpha,\beta)$-derivation if $F(xy)=F(x)\alpha(y)+\beta(x)d(y)$ holds for all $x,y\in R$, where $d$ is any mapping on $R$. Suppose that $G$ and $F$ are multiplicative (generalized)-$(\alpha,\beta)$-derivations associated with the mappings $g$ and $d$ on $R$ respectively. The main objective of this article is to study the following situations: (i) $G(xy)+F[x,y]\pm\alpha[x,y]=0$; (ii) $G(xy)+F(x\circ y)\pm\alpha(x\circ y)=0$; (iii) $G(xy)+F(x)F(y)\pm\alpha(xy)=0$; (iv) $G(xy)+F(x)F(y)\pm\alpha[x,y]=0$; (v) $G(xy)+F(x)F(y)\pm\alpha(x\circ y)=0$; for all $x,y$ in some non-zero subsets of a semiprime ring.