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!