Abstract
Let R and S be rings of a semi-projective Morita context, and alpha, beta be automorphisms of R. An additive mapping F: R -> R is called a generalized (alpha, beta) -derivation on R if there exists an (alpha, beta) -derivation d: R -> R such that F(xy) = F(x)alpha(y)+beta(x)d(y) holds for all x, y is an element of R. For any x, y is an element of R, set [x, y](alpha, beta) = x alpha(y) - beta(y)x and (x circle 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 bi-algebras via Cauchy modules.