Say I have a module $M$ over a ring $R$ that is
$1.$ finitely-generated, i.e. $\exists$ finite $X \subset M$ such that $M = \sum_{i=1}^{n} Rx_{i}, x_{i}\in X$ (but $X$ is not necessarily free)
$2.$ free, i.e. $\exists$ a basis $Y \subset M$ a.k.a. $Y$ is free and generates M, (but $Y$ is not necessarily finite).
Now I want to show, that there exists a finite basis for $M$. For vector spaces, this would obviously be trivial, but for some $R$-Modul, I'm not sure how to proceed.
Furthermore, is every minimal generating subset of $M$ a basis?
Best Answer
If the $ \beta_i $ are a basis (of unknown cardinality $ \kappa $) for the free $ R $-module $ M $, and the $ x_1, x_2, \ldots, x_n $ are a finite generating set; then each $ x_i $ is a $ R $-linear combination of finitely many of the $ \beta_i $; thus we may choose a finite subset of the $ \beta_i $ whose span includes all of the $ x_i $. But then, the span is equal to $ M $, and thus this subset is the desired finite $ R $-basis for $ M $.