If $M$ is a finitely generated abelian group, which can be made into a module over a ring $R$ (with a certain scalar multiplication) and has as a module the minimal spanning set $\{e_1, \ldots, e_n\}$, is it possible that a different ring $S$ exists, so that $M$ is a module over $S$, also with spanning set $\{e_1, \ldots, e_n\}$?
Does anybody know a non-trivial example (not $S \cong R$, $M$ not the zero module, ...)? Is it also possible for vector spaces? Thanks!
Let R be a ring with an ideal I such that R/I is finitely generated as an abelian group. Then the R/I-module R/I is minimally generated by the class of 1, and it is also an R-module minimally generated by the class of 1.