On weakly separable polynomials and weakly quasi-separable polynomials over rings

TL;DR

This paper extends the theory of weakly separable and weakly quasi-separable polynomials over rings, providing characterizations over commutative rings and necessary conditions in noncommutative skew polynomial rings.

Contribution

It offers new characterizations of weakly separable polynomials using derivatives and discriminants, and establishes conditions in noncommutative skew polynomial rings.

Findings

Characterization of weakly separable polynomials via derivatives and discriminants

Necessary and sufficient conditions for weakly separable polynomials in skew polynomial rings

Generalizations of previous results to noncommutative coefficient rings

Abstract

Separable extensions of noncommutative rings have already been studied extensively. Recently, N. Hamaguchi and A. Nakajima introduced the notions of weakly separable extensions and weakly quasi-separable extensions. They studied weakly separable polynomials and weakly quasi-separable polynomials in the case that the coefficient ring is commutative. The purpose of this paper is to give some improvements and generalizations of Hamaguchi and Nakajima's results. We shall characterize a weakly separable polynomial f(X) over a commutative ring by using its derivative f'(X) and its discriminant {\delta}(f(X)). Further, we shall try to give necessary and sufficient conditions for weakly separable polynomials in skew polynomial rings in the case that the coefficient ring is noncommutative.