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Indecomposable Almost Free Modules—The Local Case

Published online by Cambridge University Press:  20 November 2018

Rüdiger Göbel  and
Saharon Shelah
Rüdiger Göbel*
Affiliation:
Fachbereich 6, Mathematik und Informatik Universität Essen 45117 Essen Germany
Saharon Shelah*
Affiliation:
Department of Mathematics, Hebrew University, Jerusalem, Israel and Rutgers University, New Brunswick, New Jersey, U.S.A.
*
e-mail: R.Goebel@Uni-Essen.De
e-mail: Shelah@math.huji.ae.il
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Abstract

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Let $R$ be a countable, principal ideal domain which is not a field and $A$ be a countable $R$-algebra which is free as an $R$-module. Then we will construct an ${{\aleph }_{1}}$-free $R$-module $G$ of rank ${{\aleph }_{1}}$ with endomorphism algebra $\text{En}{{\text{d}}_{R}}\,G=A$. Clearly the result does not hold for fields. Recall that an $R$-module is ${{\aleph }_{1}}$-free if all its countable submodules are free, a condition closely related to Pontryagin’s theorem. This result has many consequences, depending on the algebra $A$ in use. For instance, if we choose $A\,=\,R$, then clearly $G$ is an indecomposable ‘almost free’module. The existence of such modules was unknown for rings with only finitelymany primes like $R={{\mathbb{Z}}_{\left( p \right)}}$, the integers localized at some prime $p$. The result complements a classical realization theorem of Corner’s showing that any such algebra is an endomorphism algebra of some torsionfree, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorialalgebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.

Keywords

20K2020K2620K3013C10indecomposable modules of local ringsℵ1-free modules of rank ℵ1realizing rings as endomorphism rings
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 4 , 01 August 1998 , pp. 719 - 738
Copyright
Copyright © Canadian Mathematical Society 1998

Footnotes

This work is supported by the project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development.

No. 591 from Shelah’s list of publications.

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