Indecomposable almost free modules - the local case

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=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.