Covers and envelopes with respect to a semidualizing module
type: Item
instance of: scholarly article
zbMATH Open document ID: 1242.13019
author: Xiaoguang Yan
author: Xiaosheng Zhu
publication date: 2012-06-08T00:00:00Z
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13D07
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 18G25
zbMATH DE Number: 6044051
zbMATH Keywords: semidualizing modules
zbMATH Keywords: Kaplansky class
zbMATH Keywords: Auslander class
zbMATH Keywords: preenvelope
MaRDI profile type: Publication
title: Covers and envelopes with respect to a semidualizing module
published in: Turkish Journal of Mathematics
review text: Let \(R\) be a commutative ring. Semidualizing modules were introduced by \textit{H.-B. Foxby} [Math. Scand. 31(1972), 267--284 (1973; Zbl 0272.13009)], \textit{W. V. Vasconcelos} [Divisor theory in module categories. North-Holland Mathematics Studies 14, Amsterdam-Oxford: North-Holland Publishing Comp. (1974; Zbl 0296.13005)] and \textit{E. S. Golod} [Tr. Mat. Inst. Steklova 165, 62--66 (1984; Zbl 0577.13008)] and it is a generalization of the notion of dualizing module of a Cohen-Macaulay ring. An \(R\)-module \(C\) is called semidualizing if \(C\) admits a resolution (possible infinite) by finitely generated projective modules, the natural map \(R\to \Hom_R(C,C)\) is an isomorphism, and \(\mathrm{Ext}_R^i(C,C)=0\) for \(i>0\). A class of modules \({\mathcal A}_R(C)\), the Auslander class, is associated to any semidualing module. This class consists in all the \(R\)-modules \(M\) satisfying i) \(\mathrm{Tor}_i^R(C,M)=0\) and \(\mathrm{Ext}_R^i(C,C\otimes M)=0\) for \(i>0\) and ii) the canonical map \(M\to \Hom_R(C,C\otimes_RM)\) is an isomorphism. In this paper, for a class of modules \(\mathcal Q\), a new class \({\mathcal Q}_C\) is defined by \(M\in {\mathcal Q}_C\) when \(\Hom_R(C,M)\in {\mathcal Q}\). It is shown that if \({\mathcal Q}\) is a subclass of the Auslander class which is a Kaplansky class and closed under direct sums, then the orthogonal class \({\mathcal Q}_C^\perp\) is special preenveloping. As a consequence, the class of modules of with relative to \(C\) projective dimension less or equal to \(n\) is special preenveloping. Similar results are obtained for classes contained in the Bass class associated to a semidualing module.
reviewed by: Blas Torrecillas
description: scientific article; zbMATH DE number 6044051