Skew inverse power series rings over a ring with projective socle

summary:A ring $R$ is called a right $\rm PS$-ring if its socle, ${\rm Soc}(R_{R} )$, is projective. Nicholson and Watters have shown that if $R$ is a right $\rm PS$-ring, then so are the polynomial ring $R[x]$ and power series ring $R[[x]]$. In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R[[x^{-1};\alpha , \delta ]]$ and the skew polynomial ring $R[x;\alpha , \delta ]$, where $R$ is an associative ring equipped with an automorphism $\alpha $ and an $\alpha $-derivation $\delta $. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided