Let $k$ be a commutative ring, and let $M$ and $N$ be $k$-modules. Let $\mathrm{End}(M) = \mathrm{Hom}_k (M,M)$ be the endomorphism algebra.
Is it true that $\mathrm{End}(M) \otimes \mathrm{End}(N) \cong \mathrm{End}(M \otimes N)$?
I saw this referenced on a blog but I can't find a proof of it. I tried to show that $\mathrm{End}(M \otimes N)$ satisfies the universal property of the tensor product of $\mathrm{End}(M)$ and $\mathrm{End}(N)$ but to no avail.
Thanks for any help.