ring theory – $X$ is finitely generated as a module over $A$.


$A,B$ are $k$-algebras and finitely generated as $k$-modules. Let $X$ be a simple $A otimes_k B$ module. Since $A otimes_k B$ is finitely generated as a $k$-module, $X$ is finitely generated as a $k$-module, hence also as a module over $A$ via the canonical map $A to A otimes_k B$.

I am not sure how to see that $X$ is finitely generated as a module over $A$. Any help would be appreciated!