Weakly based modules over Dedekind domains (Q397851)

A subset \(X\) of a left \(R\)- module \(M\) is weakly independent provided that whenever \(a_1 x_1+\cdots+a_n x_n=0\) for pairwise distinct elements \(x_1+\cdots+x_n\) from \(X\), then none of \(a_1,\dots,a_n\) is invertible in \(R\). Weakly independent generating sets (called 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. The authors characterize them as follows: Let \(R\) be a Dedekind domain and let \(M\) be a \(\chi\)-generated \(R\)-module, for some infinite cardinal \(\chi\). Then \(M\) has a weak basis if and only if at least one of the following conditions is satisfied:NEWLINENEWLINE (1) There are two different prime ideals \(P\), \(Q\) of \(R\) such that \(\dim_{R/P} (M/PM) = \dim_{R/Q} (M/QM) =\chi\);NEWLINENEWLINE (2) There are a prime ideal \(P\) of \(R\) and a decomposition \(M \cong F\oplus N\) where \(F\) is a free module and \(\dim_{R/P} (\tau N/P\tau N) \)= gen\((N)\);NEWLINENEWLINE (3) There is a projection of \(M\) onto an \(R\)-module \(\oplus _{P \in \mathrm{Spec}(R)} V_{p}\), where \(V_{P}\) is a vector space over \(R/P\) with \(\dim_{R/P} (V_{P}) < \chi\) for each \(P \in \mathrm{Spec}(R)\) and \(\sum_{P \in \mathrm{Spec}(R))}\dim_{(R/P} (V_{P)} = \chi\).