Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.
Question 1: Is there a nice equivalent characterisation when a ring is in $X$?
Question 2: $X$ should contain for example all Artin algebras. Does it also contain right artinian rings and if yes, is there an easy argument?
$X$ is contains semiperfect rings (hence right artinian rings) by two facts you can find in Lam. First every simple module has a projective cover. Second if $P$ is projective then $PJ\subsetneq P$ where $J$ is the radical. This latter fact is obvious if $J$ is nilpotent like for right Artinian rings and semiprimary rings but the proof for semiperfect rings is nontrivial. Any way once you know that $P/PJ$ is non-zero it has a simple quotient since $R/J$ is semisimple. That projective cover of this simple is of the form $eR$ with $e$ primitive and by definition of the projective cover, $P$ maps onto $eR$. Since this splits and $P$ is indecomposable $P\cong eR$.
I don't know more general results.
