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Transactions of the American Mathematical Society

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Representations over PID's with three distinguished submodules

Authors: Steve Files and Rüdiger Göbel
Journal: Trans. Amer. Math. Soc. 352 (2000), 2407-2427
MSC (2000): Primary 16G60, 13C05, 20K15, 20K20; Secondary 20K25, 20K40, 15A36
Published electronically: February 16, 2000
MathSciNet review: 1491863
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Let $R$ be a principal ideal domain. The $R$-representations with one distinguished submodule are classified by a theorem of Gaußin the case of finite rank, and by the ``Stacked Bases Theorem" of Cohen and Gluck in the case of infinite rank. Results of Hill and Megibben carry this classification even further. The $R$-representations with two distinguished pure submodules have recently been classified by Arnold and Dugas in the finite-rank case, and by the authors for countable rank. Although wild representation type prevails for $R$-representations with three distinguished pure submodules, an extensive category of such objects was recently classified by Arnold and Dugas. We carry their groundbreaking work further, simplifying the proofs of their main results and applying new machinery to study the structure of finite- and infinite-rank representations with two, three, and four distinguished submodules. We also apply these results to the classification of Butler groups, a class of torsion-free abelian groups that has been the focus of many investigations over the last sixteen years.

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Additional Information

Steve Files
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459

Rüdiger Göbel
Affiliation: Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany
Email: R.Goebel@Uni-Essen.DE

Keywords: Modules with distinguished submodules, decomposition into indecomposables, Butler groups, classification theorems.
Received by editor(s): November 20, 1996
Received by editor(s) in revised form: October 3, 1997
Published electronically: February 16, 2000
Additional Notes: Supported by the Graduierten Kolleg Theoretische und experimentelle Methoden der reinen Mathematik of Essen University and a project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development
Article copyright: © Copyright 2000 American Mathematical Society