1
$\begingroup$

Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$.

Can we characterize all Noetherian rings $R$ such that for every $R$-module $M$, the $R$-module $\prod_{\mathfrak p \in Spec (R)} M_{\mathfrak p}$ satisfies property $\mathcal P$ ?

$\endgroup$
1
  • $\begingroup$ By Mohan's remark, every prime ideal has to be maximal (i.e. Krull dimension is zero). In addition $R$ has to be a principal ideal ring. The property passes to finite product so we can suppose $R$ connected (hence local artinian). A few non-reduced examples exist; I'm not sure whether it's the same as artinian local PIR. $\endgroup$
    – YCor
    Nov 25, 2018 at 22:04

1 Answer 1

1
$\begingroup$

I expect that such rings are rare. If $R$ is a domain, then it must be a field. To see this, take $M=K$, the fraction field. Then $M_P=K$ for all primes $P$. But $R\subset \prod M_P$ and there are no non-zero maps from $\prod M_P\to R$ unless $R=K$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.