Is there an analogue of Bass-Papp theorem for Projective modules?

Question

The Bass-Papp theorem for injective modules states that

If $R$ is a commutative ring such that every direct sum of injective $R$ modules is injective then $R$ is Noetherian.

Is there an analogue of this theorem with the word injective replaced with projective and sum replaced by product? More explicitly is there is theorem of the type :

If $R$ is a commutative ring such that every direct product of projective $R$ modules is projective then $R$ is ________. ?

Answer 1

The theorem is due to Chase (1960) and builds on Bass (1960).

Let $R$ be a commutative, associative, unital ring. $R$ has the property that every direct product of projective modules is projective if and only if $R$ is Artinian. Chase (1960, Theorem 3.4, page 467).

Let $R$ be an associative, unital ring. $R$ has the property that every direct product of projective left modules is projective if and only if every direct power of the left regular module $R$ is left projective if and only if $R$ is left perfect and right coherent. Chase (1960, Theorem 3.3, page 467)

• Bass, Hyman “Finitistic dimension and a homological generalization of semi-primary rings.” Trans. Amer. Math. Soc. 95 (1960) 466–488. MR157984 DOI:10.2307/1993568
• Chase, Stephen U. “Direct products of modules.” Trans. Amer. Math. Soc. 97 (1960) 457–473. MR120260 DOI:10.2307/1993382