Morita context and Generalized (α, β) Derivations

Let $R$ and $S$ be rings of a semi-projective Morita context, and $alpha, eta$ be automorphisms of $R$. An additive mapping $F$: $R o R$ is called a generalized $(alpha,eta)$-derivation on $R$ if there exists an $(alpha,eta)$-derivation $d$: $R o R$ such that$F(xy)=F(x)alpha(y)+eta(x)d(y)$ holds for all $x,y in R$. For any $x,y in R$, set $[x, y]_{alpha, eta} = x alpha(y) - eta(y) x$ and $(x circ y)_{alpha, eta} = x alpha(y) + eta(y) x$. 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.