If $M$ is a finitely generated $A$-module where $A$ is Noetherian and $I$ is an ideal of $A$ such that the support of $M$, $\mathrm{Supp}(M)$ is a subset of $V(I)$ the set of prime ideals containing $I$. How do i know if there exists an $n$ such that $I^n$ is a subset of $\mathrm{Ann}(M)$?

It is relevant that if $M$ is a finitely generated module then $Supp(M)=V(\text{Ann}(M))$.

  • $\begingroup$ thank you john, i really need to start learning latex $\endgroup$ – MichaelMoore Sep 19 '15 at 20:20

Hint. $V(J)\subset V(I)\implies \mathrm{rad}(I) \subset \mathrm{rad}(J)$, so $I \subset \mathrm{rad}(J)$. Now use that $I$ is finitely generated.

  • $\begingroup$ what made you think about rad? $\endgroup$ – MichaelMoore Sep 18 '15 at 21:33
  • $\begingroup$ rad(I) is the intersection of all primes from V(I) $\endgroup$ – user26857 Sep 20 '15 at 18:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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