Modules generated by primitive idempotent elements

Question

Assume that $A$ is a finite-dimensional $k$-algebra, and $e \in A$ is a primitive idempotent element. Is it true that the submodule $eA$ is simple $A$-module? If it is, how do we prove that?

Answer 1

This statement is false in general. For example, when $A=k[x]/(x^2)$, the only idempotent it the identity element, but the left regular module (generated by $1$) is not simple.

I believe this statement is true if $A$ is semi-simple (I haven't thought through all the details, but it is definitely true when $k$ is algebraically closed). This can be derived from the Wedderburn-Artin theorem.