An alternative proof of Miyashita's theorem in a skew polynomial ring II

TL;DR

This paper provides a new, elementary proof of Miyashita's theorems regarding separable polynomials in general skew polynomial rings, extending previous results to a more comprehensive setting.

Contribution

It offers a unified, elementary proof of Miyashita's characterizations in the general skew polynomial ring with automorphism and derivation, broadening the scope of prior work.

Findings

Proofs for Miyashita's theorems in general skew polynomial rings

Extension of previous automorphism and derivation type results

Simplification of the proof methodology

Abstract

Y. Miyashita gave characterizations of a separable polynomial and a Hirata separable polynomial in skew polynomial rings. In the previous paper, the author and S. Ikehata gave direct and elementary proofs of Miyashita's theorems in skew polynomial rings of automorphism type $B[X;\rho]$ and derivation type $B[X;D]$, respectively. The purpose of this paper is to give proofs for them in the general skew polynomial ring $B[X;\rho,D]$.