Towards the decidability of the theory of modules over finite commutative rings
type: Item
instance of: scholarly article
title: Towards the decidability of the theory of modules over finite commutative rings
zbMATH Open document ID: 1168.03024
author: Gena Puninski
author: Carlo Toffalori
published in: Annals of Pure and Applied Logic
publication date: 2009-06-10T00:00:00Z
review text: The reviewer conjectured that if \(R\) is a sufficiently recursively given ring then the theory of \(R\)-modules is decidable iff \(R\) is of tame representation type. This paper begins with an excellent summary of the history of work on the problem and the current state of knowledge. It should also be recalled that the decidability problem for modules over finite rings effectively contains that for the variety generated by a finite universal algebra. The authors concentrate on the case that \(R\) is a finite commutative ring, without loss of generality local, but a good deal of their careful style of analysis applies much more generally. They make essential use of the Klingler-Levy analysis of finitely generated modules over commutative noetherian rings and they note that, in particular, this gives a clear distinction between the tame and wild cases in this context. They clearly present the interpretations involved and give a careful analysis of the information required on the Ziegler spectrum to prove decidability. They are able to establish complete results, supporting the conjecture, except in the case that \(R\) is a pullback of two finite valuation rings over a field (the Gelfand-Ponomarev algebras are long-standing examples of this open case). This is a clever and exemplary paper which might well fulfil the authors' wish to inspire further work on the problem.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 13L05
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 5563892
zbMATH Keywords: theory of modules
zbMATH Keywords: decidability
zbMATH Keywords: finite commutative ring
zbMATH Keywords: Ziegler spectrum
zbMATH Keywords: representation type
zbMATH Keywords: Klingler-Levy classification
reviewed by: Mike Prest
MaRDI profile type: Publication
full work available at URL: j.apal.2008.10.009
OpenAlex ID: W2057477641
cites work: Q5704250
cites work: Q4318498
cites work: Auslander-reiten sequences with few middle terms and applications to string algebrass
cites work: On the elementary theory of quadruples of vector spaces
cites work: The Ziegler and Zariski spectra of some domestic string algebras.
cites work: Indecomposable representations of graphs and algebras
cites work: Representations of commutative algebras
cites work: The first order properties of products of algebraic systems
cites work: Q3029166
cites work: Cancellation Counterexamples in Krull Dimension 1
cites work: Elementary Duality of Modules
cites work: The Ziegler Spectrum of a Locally Coherent Grothendieck Category
cites work: Q3999266
cites work: Representation type of commutative Noetherian rings. I: Local wildness
cites work: Representation type of commutative Noetherian rings. II: Local tameness
cites work: The structure of decidable locally finite varieties
cites work: Q3752372
cites work: Q4281263
cites work: Wild representation type and undecidability
cites work: Ziegler spectra of tame hereditary algebras
cites work: One-directed indecomposable pure injective modules over string algebras
cites work: Q4117979
cites work: Q3866236
cites work: Tame algebras and integral quadratic forms
cites work: Q4865886
cites work: The Ziegler spectrum of a tame hereditary algebra
cites work: Q4271281
cites work: Q4170856
cites work: Model theory of modules
DOI: 10.1016/J.APAL.2008.10.009
description: scientific article; zbMATH DE number 5563892