### Indecomposable almost free modules - the local case

by Goebel and Shelah. [GbSh:591]

Canadian J Math, 1998

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= 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.

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