Skew inverse power series rings over a ring with projective socle

摘要

  A ring $R$ is called a right $m PS$-ring if its socle, ${m Soc}(R_R )$, is projective. Nicholson and Watters have shown that if $R$ is a right $m PS$-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 $R[[x^{-1};lpha, delta]]$ and the skew polynomial ring $R[x;lpha, delta]$, where $R$ is an associative ring equipped with an automorphism $lpha$ and an $lpha$-derivation $delta$. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.