Some precovers and preenvelopes in functor categories
type: Item
instance of: scholarly article
title: Some precovers and preenvelopes in functor categories
zbMATH Open document ID: 1484.18001
author: Zongyang Xie
author: Zhongkui Liu
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2022-01-07T00:00:00Z
review text: Let \(R\) be an associative ring with identity, \(mod-R\) be the category of finitely presented right \(R\)-modules and \((\text{mod-}R,Ab)\) be the functor category. The authors introduce the notion of \(n\)-strongly FP-injective functor as those functors isomorphic to \(-\otimes M\) with \(M\) an \(FP_n\)-injective left \(R\)-module, i.e. Ext\(_R^1(X,M)=0\) for any finitely \(n\)-presented module \(X\), and \(n\)-strongly flat functors as those isomorphic to \((-,N)\in ((mod-R)^{op},Ab)\) with \(N\) an \(FP_n\)-flat right \(R\)-module, i.e. Tor\(^R_1(X,M)=0\) for any finitely \(n\)-presented module \(X\). In Section 3, the existence of cover and envelopes relative to this class of functors is investigated. The main result establishes that both classes of functors are preenveloping and covering in their respective categories of functors. In the last section, new characterizations of \(n\)-coherent rings [\textit{D. L. Costa}, Commun. Algebra 22, No. 10, 3997--4011 (1994; Zbl 0814.13010)] and \(n\)-regular rings [\textit{Z. Zhu}, Bull. Iran. Math. Soc. 37, No. 4, 251--257 (2012; Zbl 1277.16007)] are obtained using these classes of functors.
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 18G25
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 7453752
zbMATH Keywords: \(n\)-strongly \textit{FP}-injective functor
zbMATH Keywords: \(FP_n\)-injective
zbMATH Keywords: \(n\)-strongly flat functor
zbMATH Keywords: \(FP_n\)-flat modules
zbMATH Keywords: (pre)cover
zbMATH Keywords: (pre)envelope
reviewed by: Blas Torrecillas
MaRDI profile type: Publication
full work available at URL: s13226-021-00094-9
OpenAlex ID: W3169452124
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DOI: 10.1007/S13226-021-00094-9
description: scientific article; zbMATH DE number 7453752