Equivalent generating vectors of finitely generated modules over commutative rings

Let R be a commutative ring with identity and let M an R-module which is generated by μ elements but not fewer. We denote SLn(R) the group of n×n matrices over determinant 1. En(R) subgroup differ from single off-diagonal coefficient. Given n≥μ G∈{SLn(R),En(R)}, we study action G matrix right-multiplication on Vn(M), set Mn whose components generate M. Assuming that finitely presented elementary divisor or almost local-global coherent Prüfer ring, obtain description Vn(M)/G extends author's earlier result modules quasi-Euclidean rings [22].