How can I define the product of two ideals categorically?

Question

Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms surjections compatible, and the skeleton of this category is a partial order that can be identified with the lattice of ideals of $R$. Now, I have always been under the impression that anything one can say about ideals one can phrase in this purely arrow-theoretic language: most importantly, the intersection of ideals is the product in this category and the sum of ideals is the coproduct. (Since we're working in a partial order, product and coproduct are fancy ways to say supremum and infimum. The direction of the implied ordering on ideals may differ here from the one you're used to, but that's not important.)

However, Harry's made some comments recently that made me realize I don't know how to define the product of two ideals purely in terms of this category, that is, via a universal construction like the above. It would be really surprising to me if this were not possible, so maybe I'm missing something obvious. Does anyone know how to do this?

Answer 1

Nice question! The answer is that it's not possible! Let $R=\mathbb{F}_3[x,y]/(x^2,y^2)$. The lattice of ideals consists of the eight ideals

$(1)$

$(x,y)$

$(x)$ $(y)$ $(x+y)$ $(x-y)$

$(xy)$

$(0)$,

in which each ideal contains all ideals at lower levels. In the middle level, some of the ideals have square zero, and some don't, but you can't tell which ones just from looking at the (unlabeled) lattice.

Answer 2

Bjorn's answer shows that the product of two ideals cannot be defined by means of the partially ordered set of ideals. However, there is a category-theoretic definition of the product of two ideals if we allow us to use the category of modules. This works because of the Rosenberg reconstruction Theorem.

Explicitely, there is a bijection between ideals of $R$ and reflective, topologizing subcategories of $\text{Mod}(R)$, which is given by $I \to \{M \in \text{Mod}(R) : I M = 0\}$. If $T$ is such a subcategory with reflector $F : \text{Mod}(R) \to T$, then the corresponding ideal is $I=\ker(R \to F(R))$. Now, the multiplication of ideals corresponds to the so called Gabriel product: If $S,T$ are subcategories of an abelian category, then $S \bullet T$ is the subcategory which consists of those objects $M$ for which there is an exact sequence $0 \to M' \to M \to M'' \to 0$ with $M' \in T, M'' \in S$. The idea for the notation is $(S \bullet T)/T = S$.