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Special properties of differential inverse power series rings

Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran

E-mail Address: k.paykan@gmail.com

Corresponding author.

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A. Moussavi

Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran

E-mail Address: moussavi.a@modares.ac.ir

E-mail Address: moussavi.a@gmail.com

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https://doi.org/10.1142/S0219498816501814Cited by:11 (Source: Crossref)

Abstract

In this paper, we continue to study the differential inverse power series ring $R [[x^{- 1}; δ]]$ , where $R$ is a ring equipped with a derivation $δ$ . We characterize when $R [[x^{- 1}; δ]]$ is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, $I$ -ring, respectively. Furthermore, we prove that $R [[x^{- 1}; δ]]$ is a domain satisfying the $A C C$ on principal left ideals if and only if so does $R$ . Also, for a piecewise prime ring (PWP) $R$ we determine a large class of the differential inverse power series ring $R [[x^{- 1}; δ]]$ which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does $R [[x^{- 1}; δ]]$ . Our results extend and unify many existing results.

Communicated by André Leroy

Keywords:

AMSC: 16W60, 16W70, 16S36, 16P40

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