# Rings where all indecomposable projective modules are finitely generated

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?

• Regarding question 1, there has been quite a lot of study of a stronger condition (that every projective is a direct sum of f.g. projectives). See When every projective module is a direct sum of finitely generated modules by McGovern, Puninski and Rothmaler. But it seems quite complicated, and there are probably even more exotic examples in your class $X$: e.g., I don't see why there shouldn't be rings where not every projective is a direct sum of f.g. projectives, but there are no indecomposable projectives at all. Commented Jan 8, 2021 at 10:05

$$X$$ 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 $$P\cong eR$$.