Let $k$ be a field and $R$ an associative $k$-algebra suppose $R^2=R$. We say that $R$ is locally bounded if there exists a complete set of pairwise orthogonal primitive idempotents $\{e_x:x\in I\}$ such that $Re_x$ and $e_xR$ are finite dimensional over $k$ for all $x\in I$.
Complete set menas $R=\bigoplus_{x,y}e_x\;Re_y$. And primitive should mean that I cannot see $e_x$ as a sum of two other idempotents.
Call $P(x):= Re_x$, they are projective. Could you help me to prove that they are indecomposable? (this should follow from the fact that $e_x$ is primitive, but I cannot see why). And using Krull-Schimdt I proved that they are all the indecomposable projective modules.
Now I'm having problem with $I(x)=\mathrm{Hom}_k(e_xR,k)$. Could you tell me how to prove that they are injective, indecomposable and that they are all the injective indecomposable modules?
Finally could you tell me why $P(x)/\mathrm{rad}\;P(x)$ are all the simple modules?