Does a module (over a commutative ring) always possess a minimal generating set? When the module is not finitely generated, the typical Zorn's lemma type argument doesn't seem to work. More precisely, if $(S\_{\alpha})\_{\alpha}$ is a chain of generating subsets of a module $M$, $M=\cap \_ {\alpha}(S \_ {\alpha})\supset (\cap\_{\alpha}S\_{\alpha})$ is not in general (I think) equality (the parentheses in the last equation indicate submodule generated by).



Existence of a minimal generating set of a module
$\mathbb{Q}$, as a $\mathbb{Z}$-module, has no minimal generating set.


By the way, in the paper "A characterization of left perfect rings" by Yiqiang Zhou, it is proven that a ring $R$ is left perfect if and only if every generating set of some $R$-module contains a minimal generating set.


