On a generalization of central Armendariz rings

Document Type: Research Paper

Authors

1 Department of Mathematics, Islamic Azad University, Central Tehran Branch, 13185/768, Iran

2 Department of Mathematics and Computer Science, Shahed University, Tehran, Iran

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.‎

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