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# Indecomposable almost free modules---the local case

Published:1998-08-01
Printed: Aug 1998
• Rüdiger Göbel
• Saharon Shelah
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## Abstract

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 End$_RG = A$. 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 finitely many primes like $R = \hbox{\Bbbvii Z}_{(p)}$, 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 torsion-free, reduced $R$-module $G$ of countable rank. Its proof is based on new combinatorial-algebraic techniques related with what we call {\it rigid tree-elements\/} coming from a module generated over a forest of trees.
 Keywords: indecomposable modules of local rings, $\aleph_1$-free modules of rank $\aleph_1$, realizing rings as endomorphism rings
 MSC Classifications: 20K20 - Torsion-free groups, infinite rank 20K26 - unknown classification 20K2620K30 - Automorphisms, homomorphisms, endomorphisms, etc. 13C10 - Projective and free modules and ideals [See also 19A13]

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