Morita context and generalized (α, β)−derivations

Let $R$ and $S$ be rings of a semi-projective Morita context, and $\alpha, \beta$ be automorphisms of $R$. An additive mapping $F$: $R\to R$ is called a generalized $(\alpha,\beta)$-derivation on $R$ if there exists an $(\alpha,\beta)$-derivation $d$: $R\to R$ such that $F(xy)=F(x)\alpha(y)+\beta(x)d(y)$ holds for all $x,y \in R$. For any $x,y \in R$, set $[x, y]_{\alpha, \beta} = x \alpha(y) - \beta(y) x$ and $(x \circ y)_{\alpha, \beta} = x \alpha(y) + \beta(y) x$. In the present paper, we shall show that if the ring $S$ is reduced then it is a commutative, in a compatible way with the ring $R$ . Also, we obtain some results on bialgebras via Cauchy modules.