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Let R be a countable, principal ideal domain which is not a fleld and A be a countable R-algebra which is free as an R-module. Then we will construct an @1-free R-module G of rank @1 with endomorphism algebra EndRG = A. Clearly the result does not hold for flelds. Recall that an R-module is @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 flnitely 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.

Indecomposable Almost Free Modules—The Local Case