Let $A$ be a commutative ring and let $L$ be the set of ideals of $A$, partially ordered by inclusion. $L$ is a complete lattice but is not distributive in general. For instance, in the polynomial ring $k [x, y]$: $$((x) + (y)) \cap (x + y) = (x + y)$$ $$((x) \cap (x + y)) + ((y) \cap (x + y)) = (x (x + y), y (x + y))$$ On the other hand, sometimes $L$ is distributive – for instance, if $A$ is a principal ideal domain.
By general nonsense, there is a frame (= complete lattice in which binary meets distribute over arbitrary joins) $F$ and a "frame" homomorphism (= monotone map preserving finite meets and arbitrary joins) $L \to F$ such that every "frame" homomorphism from $L$ to a frame factors through $L \to F$ in a unique way. This is called the frame reflection of $L$.
Question. Is there a "nice" description of the frame reflection of $L$?
Again, by general nonsense, the "frame" homomorphism $L \to F$ is surjective and admits a right adjoint $F \to L$, which means we can think of $F$ as a certain subset of $L$ that is closed under arbitrary meets. In other words, $F$ is isomorphic to the poset of ideals of $A$ satisfying a certain condition – is there a "nice" description of that condition?
There is a well known frame of ideals associated with $A$, namely the frame of radical ideals of $A$, better known in its incarnation as the poset of open subsets of $\operatorname{Spec} A$. In general, $F$ is "larger" than the frame of radical ideals: for instance, if $A = \mathbb{Z}$. So $F$ corresponds to some kind of generalisation of radical ideal.
It may be worth pointing out that $A \mapsto F$ is probably not functorial. The obvious functor $A \mapsto L$ sends ring homomorphisms to monotone maps that preserve arbitrary joins, but binary meets may not be preserved. (This is basically because multiplication in the ring maps to ideal multiplication rather than ideal intersection. So we get a quantale homomorphism rather than a "frame" homomorphism – and taking the frame reflection of $L$ as a quantale yields the frame of radical ideals.)