Classification of commutative ring ideal closure operators?

Question

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?

Answer 1

Note that for any such "closure operations" one has $cl(A) = cl(\langle A\rangle)$ where $\langle A\rangle$ denotes the ideal generated by $A$. Hence it can be defined as an operation on ideals.

Now if you add a conditions of compatibility with intersection of ideals: $cl(A \cap B) = cl(A) \cap cl(B)$ for $A$ and $B$ ideals then a closure operation on a ring $R$ is the same as a nuclei on the set of ideals of $R$, i.e. it is the same as a sub-locale of the spectrum of the ring (i.e. a subspace of the spectrum from the point of view of pointfree topology).

With this insight in mind it is fairly easy to see that this has no chance to be countable: Pick any subset of prime numbers $Q$, then there is a sub-locale $Q \cup \{ \infty\}$ of $Spec \mathbb{Z}$ which will corresponds to such a closure operation on $\mathbb{Z}$:

To a set $P \subset \mathbb{Z}$ you associate the intersection of $\mathbb{Z}$ with the ideal generated by $P$ in $\mathbb{Z}[Q' ^{-1}]$ where $Q'$ is the complement of $Q$.

Now, for any ring $A$, $Spec A$ has a canonical map to $Spec \mathbb{Z}$, so you will obtain a sublocale of $Spec A$ by just pulling back any given sublocale of $Spec \mathbb{Z}$ hence any sublocale of $Spec \mathbb{Z}$ gives a canonical and functorial choices of such a closure operation on $Spec A$.

For example, here you will obtain on every ring $A$ the closure operation which associate to any set $P \subset A$ the intersection of $A$ with the ideal of $A[Q'^{-1}]$ generated by $P$.

Of course the set of subset of of prime number is uncoutable so you will obtain an uncountable number of such assignment of closure operations.

Note that if you take both the "intersection" condition and sufficiently* strong functoriality condition on your assignment in the definition you can have that a functorial assignment of closure operation is the same as a closure operation on the base ring, and hence if you chose a base ring with a finite spectrum then you can have only a finite set of such functorial assignment of closure operation.

(*) meaning that when you have a ring morphism $f :A \rightarrow B$ then the closure operation on $B$ is given by the closure operation on $A$ in the way I described above for $\mathbb{Z}$ and an aribtrary ring $A$.