Weakly based modules over Dedekind domains

We say that a subset X of a left R-module M is weakly independent provided that whenever a1x1+⋯+anxn=0 for pairwise distinct elements x1,…,xn form X, then none of a1,…,an 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 ϰ-generated R-module, for some infinite cardinal ϰ. 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 dimR/P(M/PM)=dimR/Q(M/QM)=ϰ;(2)There are a prime ideal P of R and a decomposition M≃F⊕N where F is a free module and dimR/P(τN/PτN)=gen(N);(3)There is a projection of M onto an R-module ⨁P∈Spec(R)VP, where VP is a vector space over R/P with dimR/P(VP)<ϰ for each P∈Spec(R) and ∑P∈Spec(R)dimR/P(VP)=ϰ.