Let A be an artin algebra and T, M are finitely generated right A-modules. Consider the algebra $B=End\,T$.
Why is $Hom_A(T,M)$ a finitely generated right B-module?
The action of B is clear $(f,b)\mapsto fb(t)=f(b(t))$; for $f\in Hom_A(T,M)$, $b\in B$ and $t\in T$.