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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?

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    $\begingroup$ 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 $\endgroup$
    – math54321
    Commented Apr 22, 2023 at 20:08
  • $\begingroup$ 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 $V(\mathfrak{p})$ $\endgroup$
    – math54321
    Commented Apr 22, 2023 at 20:17

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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.

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