TITLE: Example of a finitely generated module X such that End(X) is not finitely generated
QUESTION [6 upvotes]: If $R$ is a commutative Noetherian ring, then $\mathrm{Hom}_R(X,Y)$ is finitely generated $R$-module whenever $X$ and $Y$ are finite generated $R$-modules.

If $R$ is a commutative non-Noetherian ring I want find an example such that $\mathrm{End}_R(X)$ is not finitely generated $R$-module where $X$ is a finite generated $R$-module.

REPLY [4 votes]: Let $k$ be a field, $V$ an infinite dimensional vector space over $k$ and $R$ the ring with additive group $k\oplus V$and multiplication $(x,u)(y,v)=(xy,xv+yu)$, so that $V$ is an infinitely generated square zero ideal of $R$.
Take $X=R\oplus R/V$, which is clearly finitely generated.
As an $R$-module, $\text{End}_R(X)$ contains a direct summand
$$\text{Hom}_R(R/V,R)\cong V.$$