Some Results on Skew Generalized Power Series Rings

Abstract

Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $\omega \colon S \to \operatorname{End}(R)$ a monoid homomorphism. The skew generalized power series ring $R[[S,\omega]]$ is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we continue the study of skew generalized power series ring $R[[S,\omega]]$. It is shown that under suitable conditions, if $R$ has a (flat) projective socle, then so does $R[[S,\omega]]$. Necessary and sufficient conditions are obtained for $R[[S,\omega]]$ to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, clean, exchange, right stable range one, projective-free, and $I$-ring, respectively.