Suppose $R$ is an integral domain and $p$ is a prime ideal

Question

If $a\in R-p$ and $b \in p-{p}^2$, is it true that $ab \in p-{p}^2$? I can see this is obviously true in Noetherian domain but I am not sure if it is true in general (this claim about Noetherian condition was proven to be wrong by the example below).

Answer 1

Consider $R=F[x,y,z]/(xy-z^2)$ with $P=(x,z)$.

Then $y\in R\setminus P$ and $x\in P\setminus P^2$, but $xy=z^2\in P^2$.