Two criteria for locally Noetherian Grothendieck categories
type: Item
instance of: scholarly article
title: Two criteria for locally Noetherian Grothendieck categories
author: Yanyan Li
published in: Journal of Pure and Applied Algebra
publication date: 2022-09-30T00:00:00Z
review text: The author generalizes two characterizations of noetherian rings to locally noetherian Grothendieck categories. Thus, a Grothendieck category is locally noetherian if and only if every object satisfying the CS-condition is a direct sum of uniform objects. Also, a locally finitely generated Grothendieck category is locally noetherian if and only if every finitely injective object is injective if and only if every finitely injective object is a direct sum of injectives. The following result is also deduced during the process: if \(M\) is a quasi-continuous object of a locally finitely generated Grothendieck category, then \(M\) is a direct sum of indecomposable objects if and only if \(M\) has a direct sum decomposition that complements summands if and only if each local summand of \(M\) is a direct summand.
reviewed by: Septimiu Crivei
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18G05
zbMATH DE Number: 7595209
zbMATH Keywords: finitely injective objects
zbMATH Keywords: local summands
zbMATH Keywords: chain conditions
zbMATH Keywords: Grothendieck categories
MaRDI profile type: Publication
full work available at URL: j.jpaa.2022.107233
OpenAlex ID: W4293833177
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DOI: 10.1016/J.JPAA.2022.107233
description: scientific article; zbMATH DE number 7595209