Endocoherent modules.
type: Item
instance of: scholarly article
title: Endocoherent modules.
zbMATH Open document ID: 1056.16001
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: Recall that a left module \(M\) over an associative ring \(S\) with identity is coherent if it is finitely presented and every finitely generated submodule of \(M\) is finitely presented. Moreover, the module \(M\) is called \(\pi\)-coherent if it is finitely presented and every finitely generated left \(S\)-module which is cogenerated by \(M\) is finitely presented. Finally, for a right \(R\)-module \(M_R\) the symbol \(\text{Add\,}M\) (\(\text{add\,}M\)) denotes the category of all modules isomorphic to direct summands of (finite) direct sums of copies of \(M\). Let \(M_R\) be a right module over a ring \(R\) with the endomorphism ring \(S\). The following conditions are equivalent: (1) Every finitely generated left \(S\)-module which is cogenerated by \(_SM\) is finitely presented; (2) every finitely \(M\)-generated right \(R\)-module has an \(\text{add\,}M\)-preenvelope; (3) for every natural integer \(n\) and every subset \(X\subseteq M^n\) the annihilator \(\text{ann}_{S^{n\times n}}(X)\) of \(X\) in the matrix ring \(S^{n\times n}\) is a finitely generated left ideal (Theorem~1). If \(_SM\) is \(\pi\)-coherent, then every finitely generated module has an \(\text{add\,}M\)-preenvelope. The converse holds if \(M_R\) is finitely generated. If \(_SM\) is coherent, then every finitely presented module has an \(\text{add\,}M\)-preenvelope. The converse holds if \(M_R\) is finitely presented (Theorem~2).
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16P70
zbMATH DE Number: 2101118
zbMATH Keywords: \(\pi\)-coherent modules
zbMATH Keywords: categories of modules
zbMATH Keywords: preenvelopes
zbMATH Keywords: finitely presented modules
zbMATH Keywords: finitely generated modules
zbMATH Keywords: endomorphism rings
author: Lidia Angeleri Hügel
reviewed by: Ladislav Bican
MaRDI profile type: Publication
OpenAlex ID: W4240207214
DOI: 10.2140/PJM.2003.212.1
description: scientific article; zbMATH DE number 2101118