On generalizations of separable polynomials over rings

Abstract

We define that a ring extension S/R is weakly separable or weakly quasi-separable by using R-derivations of S, and give the necessary and sufficient condition that the extension R[X]/(Xⁿ − aX − b) of a commutative ring R is weakly separable. Since the notions of weakly separability and weakly quasi-separability coincide for commutative ring extensions, we treat a quotient ring R[x; *] = R[X; *]/f(X)R[X; *] of a skew polynomial ring R[X; *], and show that if R is a commutative domain, then the extension R[x; *]/R is always weakly quasi-separable, where * is either a ring automorphism or a derivation of R. We also treat the weakly separability of R[x; *]/R and give various types of examples of these extensions.