Grade of non principal Prime ideals in Noetherian UFDs

Question

I want to prove that in any Noetherian UFD the grade of every non-principal prime ideal is at least $2$.

I say in a UFD $R$ each nonzero prime ideal contains a prime element. Since the given prime ideal $P$ is not principal, it is not equal to the prime principal ideal $(p)$ which it contains, so, by virtue of $R$ being a domain, $(0)$ would be a prime ideal and therefore the height of $P$ is at least $2$. Then... ? I also thought about the fractional ideal $P^{-1}$ to be equal to $R$ to deduce that the grade of $P$ is at least $2$. Is anybody so kind as to motivate me more? Thanks in advance!

Answer 1

Let $P$ be a non-principal prime ideal, and $p \in P$ a prime element. Then $P \ne (p)$, so pick $x \in P \setminus (p)$. Can you see why $p, x$ is a regular sequence? (Note that $R/(p)$ is a domain...)