On McCoy Condition and Semicommutative Rings

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.