On a generalization of central Armendariz rings

Subject Areas : Rings theory

M. Sanaei ¹ , Sh. Sahebi ² , H. H. S. Javadi ³

1 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, 13185/768, Iran
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, 13185/768, Iran
3 - Department of Mathematics and Computer Science, Shahed University, Tehran, Iran

Received: 2018-09-01 Accepted : 2018-11-30 Published : 2019-02-01

Keywords: central Armendariz rings, $alpha$-skew Armendariz rings, central $alpha$-skew 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.‎

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