A property similar to compactly packed

Question

An ideal $I$ of a commutative ring $R$ is said to be compactly packed by prime ideals if whenever $I$ is contained in the union of a family of prime ideals of $R$, then $I$ is actually contained in one of the prime ideals of the family. Now I want to know that is there any study or mane for the property that if $I$ contains the intersection of a family of ideals of $R$, then $I$ contains one of them. (It seems to be the dual concept of compactly packed or an extention of prime ideal)

Answer 1

I did a search for "dual to compactly packed ring" and found this:

ÖZKİRİŞCİ, NESLİHAN AYŞEN, KÜRŞAT HAKAN ORAL, and ÜNSAL TEKİR. "On coprimely structured rings." Turkish Journal of Mathematics 40.4 (2016): 719-727.

In the introduction, we can find:

Gilmer formed the dual notion of compactly packed rings in his paper ”An Intersection Condition for Prime Ideals” and studied some properties of this notion [6]

Where reference [6] is

Gilmer R. An intersection condition for prime ideals. Lect Notes Pure Appl 1997; 189: 327-331.

Unfortunately I could not quickly find the text for this online, so I haven't determined what he called them yet.

Update: I finally got my hands on the Gilmer paper. His condition is:

For any nonempty family of ideals $\{I_\alpha|\alpha\in A\}$ and prime ideal $P$, $\cap_{\alpha\in A} I_\alpha\subseteq P\implies I_\beta\subseteq P$ for some $\beta$.

Gilmer showed that the commutative rings satisfying this condition for all nonempty families of ideals and all prime ideals are exactly the zero-dimensional semilocal rings (semilocal means finitely many maximal ideals.)

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Finally, I'd like to make sure you realize that this dual condition you are talking about is not an extension of prime ideals. Prime ideals satisfy this condition for finite families, but they may fail the condition for infinite families. For example, the prime ideal $\{0\}\lhd \mathbb Z$ satisfies $\bigcap\{ (p)\mid p\in\mathbb Z, p \text{ prime }\}\subseteq \{0\}$, but of course no one $(p)$ is contained in $\{0\}$.

An ideal satisfying your condition is not even necessarily prime. For example, in $\mathbb R[x]/(x^3)$, the ideal $I$ generated by $x^2$ is minimal, and obviously satisfies the condition (There are only four ideals: any intersection of ideals contained in $I$ would have to have $I$ or $\{0\}$ as one of its members.)

So, there is no axiomatic connection with prime ideals, although some additional conditions added may demonstrate some other connection.