Weakly based modules over Dedekind domains

We say that a subset X of a left R-module M is weakly independent provided that whenever a(1)x(1) + ... + a(n)x(n) = 0 for pairwise distinct elements x(1), ... , x(n) form X, then none of a(1), ... , a(n) is invertible in R. Weakly independent generating sets (we call them weak bases) are exactly generating sets minimal with respect to inclusion. The aim of the paper is to characterize modules over Dedekind domains possessing a weak basis. We will characterize them as follows: Let R be a Dedekind domain and let M be a x-generated R-module, for some infinite cardinal x. Then M has a weak basis iff at least one of the following conditions is satisfied: (1) There are two different prime ideals P, Q of R such that dim(R/P) (M/PM) = dim(R/Q) (M/QM) = x; (2) There are a prime ideal P of R and a decomposition M similar or equal to F circle plus N where F is a free module and dim(R/P) (tau N/P tau N) = gen(N); (3) There is a projection of M onto an R-module circle plus(P is an element of Spec(R)) V-P, where V-P is a vector space over R/P with dim(R/P)(V-P) < x for each P is an element of Spec(R) and Sigma(P is an element of Spec(R)) dim(R/P)(V-P)=x. (C) 2013 Elsevier Inc. All rights reserved.