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 μ elements but not fewer. We denote by SL n ( R ) the group of the n × n matrices over R with determinant 1. We denote by E n ( R ) the subgroup of SL n ( R ) generated by the matrices which differ from the identity by a single off-diagonal coefficient. Given n ≥ μ and G ∈ { SL n ( R ) , E n ( R ) } , we study the action of G by matrix right-multiplication on 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 Prüfer ring, we obtain a description of V n ( M ) / G which extends the author's earlier result on finitely generated modules over quasi-Euclidean rings [22] .