A Note on Multiplicative (generalized) (α,β)-Derivations in Prime Rings
Let $R$ be a prime ring with center $Z(R)$. A map $G\colon R\to R$ is called a multiplicative (generalized) $(\alpha,\beta)$-derivation if $G(xy) = G(x)\alpha(y)+\beta(x)g(y)$ is fulfilled for all $x,y\in R$, where $g\colon R\to R$ is any map (not necessarily derivation) and $\alpha,\beta\colon R\to R$ are automorphisms. Suppose that $G$ and $H$ are two multiplicative (generalized) $(\alpha,\beta)$-derivations associated with the mappings $g$ and $h$, respectively, on $R$ and $\alpha,\beta$ are automorphisms of $R$. The main objective of the present paper is to investigate the following algebraic identities: (i) $G(xy) + \alpha(xy) = 0$, (ii) $G(xy) + \alpha(yx) = 0$, (iii) $G(xy) + G(x)G(y) = 0$, (iv) $G(xy) = \alpha(y) \circ H(x)$ and (v) $G(xy) = [\alpha(y),H(x)]$ for all $x,y$ in an appropriate subset of $R$.(original abstract)
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