Intersection of Height One Primary Ideals is Principal

Question

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!

Answer 1

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 $(p^n)\subseteq\mathfrak q$.