Let $R$ be a ring with unity $1$ and artinian. Consider $R$ as right $R$-module $R_R$. Since it is artinian we have a finite decomposition of $R$ into right ideals:
$$R_R = A_1 \oplus ... \oplus A_n$$ where each $A_i$ is an indecomposable right ideal generated by an idempotent $e \in R$, that is $A_i = e_iR$ for every $i=1,...,n$.
I want to prove that each $e_iR$ is a local right $R$-module. By a local module I mean cyclic, non-zero and has a unique maximal proper submodule.
Do you have any idea?