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
Classification MSC:  16W60, 16W70, 16S36, 16P40
DOI:  10.21136/CMJ.2017.0672-15


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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

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