Let $M$ be an $R$-Module. Suppose we know that $M$ is finitely generated. Let $X\subseteq M$ be any generating set. Is there a finite subset of $X$ that generates $M$?
I stumbled about this when reading Eisenbud's proof of Hilbert's Basis Theorem and I'm not sure whether there is some extra hypothesis that is used at this step.
Yes, there is such a finite subset.
Because $X$ is a generating set for the module, given any $m \in M$ there exist $x_1,\dots, x_k \in X$ and scalars $r_1,\dots,r_k \in R$ such that $m = r_1 x_1 + \cdots + r_k x_k$.
Now suppose $m_1,\dots, m_n \in M$ is a finite generating set. Then for each $m_i$ there exist $x^{(i)}_1, \dots, x^{(i)}_{k(i)} \in X$ such that $m_i$ is contained in the submodule generated by those $x^{(i)}_j$. It follows that $m_1,\dots,m_n$ are all contained in the submodule generated by the finitely many $x^{(i)}_j$, so the $x^{(i)}_j$ form a finite subset of $X$ generating $M$.