ac.commutative algebra - Existence of a minimal generating set of a module

There are also simple examples of ideals of commutative rings that have no minimal generating sets, e.g., any nonzero ideal $I$ with $I = J(R)I$. (Adapt the proof of Nakayama's Lemma.) For example, let $K$ be a field, $M$ be the maximal ideal of $K[{x^s mid s in mathbb{Q}^+}]$ consisting of the elements with zero constant term, and note that $M_M = (M_M)^2$.

Since I don't have enough points to comment yet, I would also like to point out here that there is an erratum for the paper Martin referenced where the author points out that his proof is flawed. As far as I know, it is known that that condition implies perfect, but whether the converse is true is still open.