Projectivity and flatness over the endomorphism ring of a finitely generated module.
type: Item
instance of: scholarly article
title: Projectivity and flatness over the endomorphism ring of a finitely generated module.
zbMATH Open document ID: 1081.16003
DOI: 10.1155/S0161171204310082
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-18T00:00:00Z
full work available at URL: https://eudml.org/doc/52075
review text: The paper is motivated by the observation of the authors that the problem of characterizing projective (respectively flat) modules over subrings of fixed elements, or rings of invariants in specific situations [e.g. \textit{J. J. García} and \textit{A. del Río}, Math. Scand. 76, No. 2, 179-193 (1995; Zbl 0842.16023) and \textit{T. Guédénon}, Commun. Algebra 29, No. 10, 4357-4376 (2001; Zbl 0995.16030)], can be obtained from a more general, but elementary, problem of characterizing projective (respectively flat) modules over the endomorphism ring of some finitely generated (respectively finitely presented) module. Given a ring \(A\), a finitely generated right \(A\)-module \(\Lambda\) and setting \(B:=\text{End}(\Lambda_A)\), the first main result, Theorem 2.2., provides some sufficient and some necessary conditions for a right \(B\)-module \(P\) to be projective. In case \(\Lambda_A\) is semi \(\Sigma\)-projective (in particular quasi-projective), it is shown that \(P_B\) is projective if and only if \(P\simeq\Hom_A(\Lambda,Q)\) as right \(B\)-modules for some direct summand \(Q\subseteq_\oplus\Lambda^{(I)}\). Given a ring \(A\), a finitely presented right \(A\)-module \(\Lambda\) and setting \(B:=\text{End}(\Lambda_A)\), the second main result, Theorem 2.3., provides characterizations of a right \(B\)-module \(P\) to be flat. In particular it is shown that \(P_B\) is flat if and only if \(P\simeq\Hom_A(\Lambda,Q)\) as right \(B\)-modules, where \(Q\simeq\varinjlim\Lambda^{n_i}\) for some positive integers \(\{n_i\}_I\). An interesting application in the paper is Example 3.3.: let \(\Lambda\) be a ring, \(({\mathcal C},\Delta,\varepsilon)\) be a \(\Lambda\)-coring with a group-like element \(x\) and consider the left dual ring \(^*{\mathcal C}:={_\Lambda\Hom}({\mathcal C},\Lambda)\) with the convolution product. Then \(\Lambda\) becomes a cyclic \(^*{\mathcal C}\)-module through \(\lambda\leftharpoonup f:=f(x\lambda)\) for all \(\lambda\in\Lambda\) and \(f\in{^*{\ma…
reviewed by: Jawad Abuhlail
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W22
zbMATH DE Number: 2168709
zbMATH Keywords: endomorphism rings
zbMATH Keywords: projective modules
zbMATH Keywords: flat modules
zbMATH Keywords: subrings of invariants
zbMATH Keywords: group-like elements
zbMATH Keywords: corings
zbMATH Keywords: finitely generated modules
zbMATH Keywords: finitely presented modules
author: Stefaan Caenepeel
author: T. Guédénon
MaRDI profile type: Publication
OpenAlex ID: W2117119545
description: scientific article; zbMATH DE number 2168709