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 $R$ ? If that is not true in general, then what if we also assume $R$ has finite dimension, or say very low dimension (say 1) ?

The Noetherian case obvious from Krull Intersection theorem, and so is the case when $R$ is s Prufer domain , due to Corollary 2.4 (a) of https://cms.math.ca/openaccess/cjm/v18/cjm1966v18.1024-1030.pdf . So any possible counterexample would have to be non-Noetherian and non-Prufer.

The Krull intersection theorem,Pacific J. Math. 57 (1975), 11-14. $\endgroup$ – Fred Rohrer Jun 26 '18 at 12:27