It is well-known that a direct sum of modules is projective if and only if each summand is projective, and a direct product of modules is injective if and only if each of the modules are injective. Also, by a thorem of Bass, that any direct sum of injective right (respectively left) $R$-modules is injective is equivalent to right (respectively left) Noetherian-ness of the ring $R$. My question is that is there any sufficient, or necessary (or both) condition about projectiveness of any direct product of projective $R$-modules? I would appreciate anyone who has information thereof and communicating with me.
2026-03-28 22:28:32.1774736912
Projectiveness of direct product of projectives
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All direct products of projective right $R$ modules are projective iff $R$ is right perfect and left coherent.
I believe it appears in Lam's Lectures on rings and modules or Anderson & Fuller's book (or both).
If you know the theorem that all products of flat right modules are flat iff a ring is left coherent, and that the flat and projective modules coincide over a right perfect ring, then you have a proof that way too.