# Sh:591

• Göbel, R., & Shelah, S. (1998). Indecomposable almost free modules—the local case. Canad. J. Math., 50(4), 719–738.
• 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={\mathbb 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 rigid tree-elements coming from a module generated over a forest of trees.
• Version 2000-10-31_11 (25p) published version (20p)
Bib entry
@article{Sh:591,
author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon},
title = {{Indecomposable almost free modules---the local case}},
}