Equivalent generating vectors of finitely generated modules over commutative rings

Abstract

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module which is generated by $\mu$ elements but not fewer. We denote by $\operatorname{SL}_n(R)$ the group of the $n \times n$ matrices over $R$ with determinant $1$. We denote by $\operatorname{E}_n(R)$ the subgroup of $\operatorname{SL}_n(R)$ generated by the the matrices which differ from the identity by a single off-diagonal coefficient. Given $n \ge \mu$ and $G \in \left\{\operatorname{SL}_n(R),\operatorname{E}_n(R)\right\}$, we study the action of $G$ by matrix right-multiplication on $\operatorname{V}_n(M)$, the set of elements of $M^n$ whose components generate $M$. Assuming that $M$ is finitely presented and that $R$ is an elementary divisor ring or an almost local-global coherent Pr\"ufer ring, we obtain a description of $\operatorname{V}_n(M)/G$ which extends the author's earlier result on finitely generated modules over quasi-Euclidean rings.