On partial skew Armendariz rings

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.