# When is prime $\mathfrak p$ the intersection of all proper primes containing it?

I am looking for a name/reference to this property(I call it reachable for now)

A prime ideal $$\mathfrak p\subset A$$ is called reachable if it is the intersection of all proper prime ideals containing it, i.e., $$\mathfrak p=\bigcap_{\mathfrak q\supset \mathfrak p, \mathfrak q\neq \mathfrak p}\mathfrak q$$

Here is a highly non-trivial result of interest:

Let $$f: A\to B$$ be a finite type ring map, and $$\mathfrak p\subset B$$ is a prime ideal. If $$f^{-1}(\mathfrak p)$$ is reachable, then $$\mathfrak p$$ is also reachable.

For a ring map, preserving reachable primes is weaker than going-up, but going up definitely fails for a finite type ring map. Usually we require it to be intergal.

Is there a name/some reference to this property of interest?

• You should re-define reachable to automatically include maximal ideals. With this modification, every prime is reachable iff the ring is Jacobson. For a Noetherian ring, any prime of dimension $\ge 2$ is always reachable. In particular, TFAE for a Noetherian ring $R$: (i) $R$ is Jacobson, (ii) every prime of dim $1$ is reachable, (iii) every prime of dim $1$ is contained in infinitely many maximal ideals Commented Apr 22, 2023 at 20:08
• Another general fact: a prime $\mathfrak{p}$ is not reachable iff there exists $f \in R$ such that the localization $(R/\mathfrak{p})_f$ is a field. Equivalently, $\{ \mathfrak{p} \}$ is open in $V(\mathfrak{p})$ Commented Apr 22, 2023 at 20:17

To put it in terms of something that exists in the literature, $$\mathfrak {p}$$ is reachable iff it is not a Goldman ideal. See this post to see why $$R/\mathfrak{p}$$ is not Goldman, and the wiki.