TITLE: Rings where all indecomposable projective modules are finitely generated
QUESTION [10 upvotes]: 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?

REPLY [5 votes]: $X$ 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.