α-Skew π-McCoy Rings

Abstract

As a generalization of $\alpha$-skew McCoy rings, we introduce the concept of $\alpha$-skew $\pi$-McCoy rings, and we study the relationships with another two new generalizations, $\alpha$-skew ${\pi }_1$-McCoy rings and $\alpha$-skew ${\pi }_2$-McCoy rings, observing the relations with $\alpha$-skew McCoy rings, $\pi$-McCoy rings, $\alpha$-skew Armendariz rings, $\pi$-regular rings, and other kinds of rings. Also, we investigate conditions such that $\alpha$-skew ${\pi }_1$-McCoy rings imply $\alpha$-skew $\pi$-McCoy rings and $\alpha$-skew ${\pi }_2$-McCoy rings. We show that in the case where $R$ is a nonreduced ring, if $R$ is 2-primal, then $R$ is an $\alpha$-skew $\pi$-McCoy ring. And, let $R$ be a weak ($\alpha ,\delta$)-compatible ring; if $R$ is an $\alpha$-skew ${\pi }_1$-McCoy ring, then $R$ is $\alpha$-skew ${\pi }_2$-McCoy.