Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 389-395, 2017
Skew inverse power series rings over a ring with projective socle
Kamal Paykan
Received December 11, 2015. First published March 20, 2017.
Abstract: 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.
Keywords: skew inverse power series ring; skew polynomial ring; annihilator; projective socle ring; flat socle ring
Affiliations: Kamal Paykan, Department of Basic Sciences, Garmsār Branch, Islamic Azad University, 3581631167 Garmsār, Iran, e-mail: k.paykan@gmail.com, k.paykan@modares.ac.ir