On a generalization of central Armendariz rings

Abstract

In this paper, some properties of $alpha$-skew Armendariz and central Armendariz rings have been studied by variety of others. We generalize the notions to central $alpha$-skew Armendariz rings and investigate their properties. Also, we show that if $alpha(e)=e$ for each idempotent $e^{2}=e in R$ and $R$ is $alpha$-skew Armendariz, then $R$ is abelian. Moreover, if $R$ is central $alpha$-skew Armendariz, then $R$ is right p.p-ring if and only if $R[x;alpha]$ is right p.p-ring. Then it is proved that if $alpha^{t}=I_{R}$ for some positive integer $t$, $ R $ is central $ alpha $-skew Armendariz if and only if the polynomial ring $ R[x] $ is central $ alpha $-skew Armendariz if and only if the Laurent polynomial ring $R[x,x^{-1}]$ is central $alpha$-skew Armendariz.‎