Finitely generated modules over pullback rings
type: Item
instance of: scholarly article
title: Finitely generated modules over pullback rings
zbMATH Open document ID: 0863.16003
author: Reinhard Laubenbacher
author: David M. Arnold
published in: Journal of Algebra
publication date: 1997-06-09T00:00:00Z
review text: The completion of the algebra \(k[x,y:xy=0]\) (\(k\) is a field) at its maximal ideal \(\langle x,y\rangle\) may be obtained as the pullback of two valuation domains mapping to a common factor field. \textit{L. Levy} [J. Algebra 93, 1-116 (1985; Zbl 0564.13010)] classified the finitely generated modules over such pullback (or, more generally, Dedekind-like) rings. In this paper the authors partly generalise Levy's results (to rings which are pullbacks of two hereditary noetherian homogeneously serial rings, finitely generated over their centres, mapping to a common semisimple artinian factor ring). Their approach is to analyse separately the maximal artinian submodule, \(A(M)\), and the quotient, \(M/A(M)\), of an indecomposable finitely generated module \(M\) (they show that these determine \(M\) among indecomposable modules). Their analysis goes via the associated graded ring of the pullback ring and, in particular, their analysis of the artinian modules uses the ``string and band'' classification [see, e.g., \textit{M. C. R. Butler} and \textit{C. M. Ringel}, Commun. Algebra 15, 145-179 (1987; Zbl 0612.16013)] of finite length modules over the first-mentioned algebra. A couple of typos: on p. 315 line 8, read ``\(JA\)'' not ``\(IA\)'' and, on the next line, insert ``homogeneously'' before the second occurrence of ``serial''.
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 934255
zbMATH Keywords: pullback of valuation domains
zbMATH Keywords: pullback rings
zbMATH Keywords: completions
zbMATH Keywords: finitely generated modules
zbMATH Keywords: hereditary Noetherian homogeneously serial rings
zbMATH Keywords: maximal Artinian submodules
zbMATH Keywords: indecomposable finitely generated modules
zbMATH Keywords: finite length modules
reviewed by: Mike Prest
MaRDI profile type: Publication
OpenAlex ID: W2007828801
DOI: 10.1006/JABR.1996.0262
description: scientific article; zbMATH DE number 934255