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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus


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Online
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1435-4446
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Volume 10, Issue 4

Issues

Constructing sharply transitive R-modules of rank ⩽

Rüdiger Göbel / Daniel Herden
Published Online: 2007-07-26 | DOI: https://doi.org/10.1515/JGT.2007.039

Abstract

In this note we will give an elementary proof of the existence of sharply transitive R-modules M over principal ideal domains R. An R-module is sharply transitive (or a UT-module) if its R-automorphism group acts sharply transitively on the pure elements of M. We will assume that M is torsion-free; thus pure elements are simply those elements divisible only by units of R in M. We provide examples of UT-modules of rank ⩽ , while the existence of UT-modules of rank ⩾ was shown recently in Göbel and Shelah [R. Göbel and S. Shelah. Uniquely transitive torsion-free abelian groups. In Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Applied Math. 236 (Marcel Dekker, 2004), pp. 271–290.] using the more complicated machinery of prediction principles. The existence of countable abelian UT-groups, which follows from this note, was left open in earlier works. Here we require and exploit the existence of algebraically independent elements over the base ring R. (Thus we will need |R| < .) First we will convert the UT problem on modules (as suggested in Herden [D. Herden. Uniquely transitive R-modules. Ph.D. thesis. University of Duisburg-Essen, Campus Essen (2005).]) into a problem on suitable R-algebras. This reduces its solution to a few simple steps and makes the proofs more transparent, requiring only basic results in module theory.

About the article


Received: 2006-04-04

Revised: 2006-09-19

Published Online: 2007-07-26

Published in Print: 2007-07-20


Citation Information: Journal of Group Theory, Volume 10, Issue 4, Pages 467–475, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/JGT.2007.039.

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Citing Articles

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[1]
Rüdiger Göbel
Journal of Pure and Applied Algebra, 2011, Volume 215, Number 5, Page 822
[2]
Rüdiger Göbel, Daniel Herden, and Saharon Shelah
Advances in Mathematics, 2011, Volume 226, Number 1, Page 235
[3]
Daniel Herden and Saharon Shelah
Forum Mathematicum, 2010, Volume 22, Number 4, Page 627

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