PF-rings of skew generalized power series

Abstract

Let $R$ be a ring which is $S$-compatible and $(S,\omega)$-Armendariz. In this paper, we investigate that the skew generalized power series ring $R[[S,\omega]]$ is a PF-ring if and only if for any two $S$-indexed subsets $P$ and $Q$ of $R$ such that $Q \subseteq ann_R (P)$ and there exists $a\in ann_R (P)$ such that $q a=q$ for all $q \in Q$. Further, we prove that if $R$ be a Noetherian ring then $R[[S,\omega]]$ is a PP-ring if and only if $R$ is a PP-ring.