On partial skew Armendariz rings

Abstract

In this paper we consider rings \(R\) with a partial action \(\alpha\) of an infinite cyclic group \(G\) on \(R\). We introduce the concept of partial skew Armendariz rings and partial \(\alpha\)-rigid rings. We  show that partial \(\alpha\)-rigid rings are  partial skew Armendariz rings and  we give necessary and sufficient conditions for \(R\) to be a partial skew Armendariz ring. We study the transfer of Baer property, a.c.c.  on right annhilators property, right p.p. property and right zip property between \(R\) and \(R[x;\alpha]\).

We also show that  \(R[x;\alpha]\) and \(R\langle x;\alpha\rangle\) are not necessarily associative rings when \(R\) satisfies  the concepts mentioned above.