As I learned here, since the very beginning of humanity/Kummer's development of ideals, the idea that they are "numbers" that we can "modulo by"/do "modular arithmetic by" has been central to the concept. An example that was given in the linked post is that the ideal $(2,1+\sqrt{-5})$ in $\mathbb Z[\sqrt{-5}]$ can be thought of as (the kernel of) the ring homomorphism $$f:\mathbb Z[\sqrt{-5}] \to \mathbb Z/2\mathbb Z, f(a+b\sqrt{-5}) := a+b+2{\mathbb Z}.$$
It took until the work of Dedekind (and even then, the later revisions) that the notion of ideals as sets of elements of a ring took center stage, especially in the seemingly-naive but miraculously-suitable definition of multiplying 2 ideals. (Indeed, somehow all the stars align and we get unique factorization of ideals in terms of the aforementioned definition of product ideal in, for instance, all number rings. Although using Kummer's ring isomorphism conception of ideals we can talk about ideals dividing/divisible by other ideals, we seem to be missing the notion of multiplicity of "how many times" an ideal divides another — perhaps this is some intuition for the alternative definition of Dedekind domain: Noetherian domain s.t. all localizations at prime ideals are DVRs, where discrete valuations are exactly the framework for talking about multiplicities [and Noetherianity is there to guarantee finiteness of the prime factorization process]).
Question: I'm wondering if there's any way to interpret the multiplication of ideals "in Kummer's original conception of the ideal", or more concretely, is there somehow a way to construct, using ring homomrphisms that define $I_1,I_2$, a ring homomorphism whose kernel is $I_1\cdot I_2$?
I'm aware that the Chinese remainder theorem does this if $I_1,I_2$ are relatively prime, but it doesn't seem to apply in general.
Best Answer
I don't believe there is a way to do this. The natural thing to do with two quotients $R \to R/I_1$ and $R \to R/I_2$ is to consider their product $R \to R/I_1 \times R/I_2$, and of course what this produces is the intersection.
Years ago I asked a related question on MathOverflow about whether the ideal product was some categorical operation on the poset of ideals (the intersection and the sum are the meet and the join respectively), and the answer turns out to be no.
Honestly I've always thought of the ideal product as sort of an anomaly for this reason. The categorically natural operations are the intersection, the sum, and the tensor product, and I don't think the ideal product is a particularly natural thing to look at unless it agrees with one of these. (In the case of fractional ideals of a Dedekind domain $D$, ideal product agrees with the tensor product, at least up to isomorphism of modules; in fact the ideal class group is naturally isomorphic to the Picard group of line bundles on $\text{Spec } D$, with group operation the tensor product.)