INDECOMPOSABLE ALMOST FREE MODULES - THE LOCAL CASE

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 N-1-free a-module G of rank N-1 with endomorphism algebra End(R) G = A. Clearly the result does not hold for fields. Recall tha t an R-module is N-1-free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has man y consequences, depending on the algebra A in use. For instance, if we choose A = R, then clearly G is an indecomposable 'almost free' modul e. The existence of such modules was unknown for rings with only finit ely 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 torsi onfree, reduced R-module G of countable rank. Its proof is based on ne w combinatorial-algebraic techniques related with what we call rigid t ree-elements coming from a module generated over a forest of trees.