# Sh:591

- Göbel, R., & Shelah, S. (1998).
*Indecomposable almost free modules—the local case*. Canad. J. Math.,**50**(4), 719–738. arXiv: math/0011182 DOI: 10.4153/CJM-1998-039-7 MR: 1638607 -
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. - 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}}, journal = {Canad. J. Math.}, fjournal = {Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques}, volume = {50}, number = {4}, year = {1998}, pages = {719--738}, issn = {0008-414X}, doi = {10.4153/CJM-1998-039-7}, mrclass = {13C13 (20K20)}, mrnumber = {1638607}, mrreviewer = {Luigi Salce}, doi = {10.4153/CJM-1998-039-7}, note = {\href{https://arxiv.org/abs/math/0011182}{arXiv: math/0011182}}, arxiv_number = {math/0011182} }