First, some setup:

So: given a commutative ring $R$, let $Ideals(R)$ be set of ideals of $R$ and let $IdealClosure(R)$ be the set of closure operations $cl: \mathcal{P}(R) \rightarrow Ideals(R)$. In other words, let $IdealClosure(R)$ be the set of functions $cl: \mathcal{P}(R) \rightarrow \mathcal{P}(R)$ such that

- $cl(A) \subseteq R$ is an ideal for all $A \subseteq R$,
- $A \subset cl(A)$ for all $A \subseteq R$,
- $cl(cl(A)) = cl(A)$ for all $A \subseteq R$, and
- $A_1 \subseteq A_2 \implies cl(A_1) \subseteq cl(A_2)$ for all $A_1,A_2 \subseteq R$.

Note that since the ideals of a ring form a bounded lattice, the set $IdealClosure(R)$ is also a bounded lattice. Now if $\mathcal{R}$ is some set of rings, let $IdealClosure(\mathcal{R})$ be the collection of functions which assign to each ring $R$ an element of $IdealClosure(R)$. Again, $IdealClosure(\mathcal{R})$ is a bounded lattice by defining the lattice operations pointwise, ring by ring.

Let $\mathcal{R}$ be a set containing at least one representative from each isomorphism class of the finitely presented commutative rings. Then we can assume $\mathcal{R}$ is countable.

It seems then that given a few more 'naturality' conditions the set $IdealClosure(\mathcal{R})$ would likewise countable. For example, without going to far into symbolic logic, a natural ideal closure operator would be defined by one of a countable collection of statements in FOL or ZFC.

One obvious condition is this: an ideal closure operation should treat isomorphic rings equivalently, or more formally: if $R_1 \in \mathcal{R}$, $R_2 \in \mathcal{R}$, $\varphi: R_1 \rightarrow R_2$ is an isomorphism, and $cl_1 = c(R_1)$, $cl_2 = c(R_2)$ where $c \in IdealClosure(\mathcal{R})$ then $\varphi[cl_1(A)] = cl_2[\varphi(A)]$ for all $A \subseteq R$.

What other conditions are reasonable? Are such sets of closure operations studied?