On the existence of non-trivial finitely injective modules.
type: Item
instance of: scholarly article
title: On the existence of non-trivial finitely injective modules.
zbMATH Open document ID: 1319.16004
author: Pedro A. Guil Asensio
author: Manuel C. Izurdiaga
author: Blas Torrecillas
published in: Forum Mathematicum
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.3926
reviewed by: Samir Bouchiba
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 6368604
zbMATH Keywords: finitely injective modules
zbMATH Keywords: left Noetherian rings
zbMATH Keywords: direct sums
Wikidata QID: Q114053049
MaRDI profile type: Publication
OpenAlex ID: W2335696303
arXiv ID: 1204.3926
DOI: 10.1515/FORUM-2012-0045
review text: The main goal of the present paper is to characterize left Noetherian rings in terms of the existence of non-trivial finitely injective modules. Besides, the authors provide a simple way to construct non-trivial finitely injective modules.NEWLINENEWLINE Let \(R\) be a ring. Recall that a module \(M\) over \(R\) is said to be finitely injective if each finite subset of \(M\) is contained in an injective submodule of \(M\). Any direct sum of injective modules is trivially finitely injective. Thus, a finitely injective \(R\)-module \(M\) is said to be trivial if it is a direct sum of injective \(R\)-modules. This paper gives a positive answer to an open question of L. Salce by proving that a ring \(R\) is left Noetherian if and only if each finitely injective left \(R\)-module is trivial.
description: scientific article; zbMATH DE number 6368604