Abstract

We investigate the Gorenstein semihereditary rings and Gorenstein Prufer domains in terms of the notion of the copure flat dimension $cfD(R)$ of a ring $R$ which is defined in [X. H. Fu and N. Q. Ding, Comm. Algebra, 38(12) (2010), 4531-4544].

Highlights

  • Throughout this paper, R is an associative commutative ring with identity

  • A ring R is said to be hereditary if every ideal of R is projective, and a hereditary domain is called a Dedekind domain

  • The copure projective dimension of a ring R is defined as cpD(R) = sup{ cpdR(M ) | M is an R-module }

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Summary

Introduction

Throughout this paper, R is an associative commutative ring with identity. For an R-module M , fdRM (resp. idRM ) stands for the flat (resp. injective) dimension of M. The copure projective dimension of a ring R is defined as cpD(R) = sup{ cpdR(M ) | M is an R-module }. In [33,35], Xiong et al proved that a ring R has cpD(R) ≤ 1 if and only if every submodule of a projective R-module is copure projective In this case, R is said to be a CPH (Copure-Projective-Hereditary) ring provisionally. R is said to be a CPH (Copure-Projective-Hereditary) ring provisionally They proved that a domain R is a Gorenstein Dedekind domain if and only if cpD(R) ≤ 1. In the paper [6] the author defined the copure flat dimension cf dRM of an Rmodule M to be the largest integer n ≥ 0 such that TorRn (E, M ) = 0 for some injective R-module E. In terms of this result, we study the Gorenstein Prufer domains

Semi-CPH rings and Gorenstein semihereditary rings
Gorenstein Prufer domains
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