Suppose we have two finitely generated modules $M$ and $N$ over a Noetherian ring $R$. If $M\otimes_RN$ is a nonzero projective $R$-module (or even better, a nonzero free $R$-module), can we conclude that both $M$ and $N$ are projective $R$-modules? If not, what condition should we put on $R$ to guarantee that?
Asked
Active
Viewed 168 times
3
-
Nice question! See https://math.stackexchange.com/questions/733522/if-the-tensor-product-of-two-modules-is-free-of-finite-rank-then-the-modules-ar for an affirmative answer to the second question. Not sure about the projective case, though I think the second answer to the linked question answers this as well. – Alex Wertheim Apr 03 '23 at 03:41
-
@AlexWertheim Thanks! It does help! – S.Gau at Math Apr 03 '23 at 22:53