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).
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$\mathbb{Q}$, 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. 

