# Intersection of all positive powers of a prime ideal in integral domain with all ideals of finite height

Let $R$ be an integral domain with every prime ideal having finite height. Then is $\bigcap_{n>1} P^n$ a prime ideal of $R$ for every prime ideal $P$ of $R$ ?

The Noetherian case obvious from Krull Intersection theorem, so any possible counterexample would have to be non-Noetherian.

## 1 Answer

David Speyer's example of $$R = \bigcup_{n=1}^{\infty} k\left[x,\ y,\ x^{1/n!} y^{1/n!} \right]$$ for any field $$k$$ also works for this question. The ideal $$P=(x, y, x y, x^{1/2} y^{1/2}, x^{1/3} y^{1/3}, x^{1/4} y^{1/4}, \cdots )$$ is prime but $$\bigcap P^n$$ is not prime (see the linked answer for details).

Moreover, I claim every prime ideal in $$R$$ has height at most $$2$$. Indeed, suppose $$Q_0\subset Q_1\subset Q_2\subset Q_3$$ is a chain of prime ideals in $$R$$. There is then some $$n$$ such that this chain of inclusions remains strict when restricted to $$R_n=k[x,y,x^{1/n!}y^{1/n!}]$$ (since $$R$$ is the direct limit of these subrings $$R_n$$). But $$R_n$$ has dimension $$2$$ (it is an integral extension of $$k[x,y]$$), so this is impossible.