Classification theory for abelian groups with an endomorphism
type: Item
instance of: scholarly article
title: Classification theory for abelian groups with an endomorphism
zbMATH Open document ID: 0723.03020
DOI: 10.1007/BF01387762
author: Annalisa Marcja
author: Carlo Toffalori
author: Mike Prest
published in: Archive for Mathematical Logic
publication date: 1991-01-01T00:00:00Z
review text: The paper considers classifiability for Abelian groups with an endomorphism x, hence for Z[x]-modules. In stability theory, classifying the models of some complete theory means to determine their isomorphism types by some kind of invariant (like a ``short'' sequence of cardinals). When the models are modules over some fixed countable ring R, this assignment of invariants can be done if and only if the theory is superstable. Moreover there are good reasons to believe that, if the ring R is of wild representation type, then this classification is strongly complicated, even if not compromised. The authors first provide a full characterization of superstable K[x]- modules where K is any countable field (notice that this includes the case \(K={\mathbb{Q}}\), hence Z[x]-modules which are divisible and torsionfree over Z). Secondly they consider Z[x]-modules whose underlying group has a finite exponent m, hence Z/m[x]-modules. Here it is shown that in most cases Z/m[x] and its quotient rings are of wild representation type, and the corresponding theories of modules are undecidable. The same happens for Z[x]-modules which are divisible (or torsionfree) as Z-modules. A parallel analysis is given when x is an automorphism, so for \(Z[x,x^{- 1}]\)-modules.
reviewed by: Carlo Toffalori
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20E99
zbMATH DE Number: 4191608
zbMATH Keywords: Abelian groups with an endomorphism
zbMATH Keywords: classification
zbMATH Keywords: superstable K[x]- modules
zbMATH Keywords: Z[x]-modules
zbMATH Keywords: wild representation type
zbMATH Keywords: automorphism
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01387762
OpenAlex ID: W2076105251
description: scientific article; zbMATH DE number 4191608