On McCoy Condition and Semicommutative Rings

TL;DR

This paper generalizes McCoy's Theorem for skew polynomial rings, establishing conditions under which the right annihilator of an ideal in the extension implies the same in the base ring, and explores semicommutativity in these contexts.

Contribution

It extends McCoy's Theorem to skew polynomial rings with $\sigma$-stability or compatibility, and characterizes semicommutativity and McCoy properties in Nagata extensions.

Findings

If $r_S(I)$ is $\sigma$-stable or compatible, then $r_S(I) eq 0$ implies $r_R(I) eq 0$.

$R[x;\sigma]$ semicommutative implies $R$ is $\sigma$-skew McCoy.

Nagata extension $R igoplus_{\sigma} M_R$ is semicommutative and McCoy when $R$ is a commutative domain.

Abstract

Let $R$ be a ring, $\sigma$ an endomorphism of $R$, $I$ a right ideal in $S=R[x;\sigma]$ and $M_R$ a right $R$-module. We give a generalization of McCoy's Theorem \cite{mccoy}, by showing that, if $r_S(I)$ is $\sigma$-stable or $\sigma$-compatible. Then $\;r_S(I)\neq 0$ implies $r_R(I)\neq 0$. As a consequence, if $R[x;\sigma]$ is semicommutative then $R$ is $\sigma$-skew McCoy. Moreover, we show that the Nagata extension $R\oplus_{\sigma}M_R$ is semicommutative right McCoy when $R$ is a commutative domain.