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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?

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    $\begingroup$ 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. $\endgroup$ Jan 8, 2021 at 10:05

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$X$ 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.

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