Suppose $R$ is Noetherian and $M$ is a finitely generated $R$-module. IF $g \in \text {End}_R(M,M)$ does there exist a k such that $g^k =g^{k+1}= \ldots$
I'm trying to work with the generators, but not getting anywhere. Any assistance would be appreciated.
No, not in general. Let $R=\mathbb{K}$ be a field and let $M$ be a vector space of dimension $2$ over $\mathbb{K}$. Write $M=\mathbb{K}e_1\oplus\mathbb{K}e_2$. Then the map $g$ defined by $e_1 \mapsto e_2$ and $e_2\mapsto e_1$ has $g^k\neq g^{k+1}$ for every $k$.