Let $R$ be a UFD having height one primary ideals $\mathfrak q_1,...,\mathfrak q_r$. My purpose is to show that their intersection is principal.

I do know that in a UFD each prime ideal of height one is principal.

Thanks for any help!


A height one primary ideal in a UFD is principal and generated by a power of a prime.

Let $\mathfrak q$ be a $\mathfrak p$-primary ideal of height one. Then $\mathfrak p=(p)$ with $p\in R$ a prime element. We have $\mathfrak q\subseteq (p)$. Suppose $\mathfrak q\subseteq (p^n)$, and $\mathfrak q\nsubseteq(p^{n+1})$. Since $\sqrt{\mathfrak q}=(p)$, there is $k\ge 1$ such that $p^k\in\mathfrak q$. Then $k=n$, so $(p^n)\subseteq\mathfrak q$.

| cite | improve this answer | |
  • 1
    $\begingroup$ What happens if $q \subset \cap_{n \geq 0}(p^{n}) ?$ I mean why it cannot happen ? $\endgroup$ – user371231 Nov 7 '18 at 7:48
  • $\begingroup$ @user371231 In a UFD which elements are divisible by $p^n$, for every $n\ge 1$? $\endgroup$ – user26857 Nov 7 '18 at 17:57
  • 1
    $\begingroup$ Ok, thank you. It is elementary but seems to me very useful. $\endgroup$ – user371231 Nov 7 '18 at 19:45

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.