I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. You can define rings and modules similarly. You can define a field object as a commutative ring object with the inverse axiom from a group object applied to the multiplication (alternately as a commutative ring object over which every module is free). Assuming your category is abelian, one would hope that you could define a maximal ideal $M$ by an exact sequence $0 \rightarrow M \rightarrow R \rightarrow k \rightarrow 0$ where $k$ is a field object. I'm thinking ahead to generalizing this to a triangulated category where $M$ could be uniquely defined via a fiber sequence.
I’d like to define an ideal $I$ as a subobject of $R$ which is also an $R$-bimodule. Both of these can be defined by diagrams, without reference to elements. However, I'm a bit afraid that this is completely wrong since I've never heard of an "ideal object", and it seems Google has not either. Is there something I’m missing here that makes my definition fail?
The next object I’d like to define is a prime ideal. I have no idea how to do this without referring to elements. The trick with maximal ideals above doesn’t seem to work unless someone has a way to classify “integral domain objects” via diagrams.
Has anyone ever heard of a way to define prime ideals purely diagrammatically?
Alternately, assuming that I totally botched that attempt to define an "ideal object," one could still ask about prime modules, i.e. nonzero modules $M$ such that $IN = (0)$ implies $N = (0)$ or $IM = (0)$ for any ideal $I$ of $R$ and any submodule $N$ of $M$. Has anyone heard of a diagrammatic way to define these?
I'm tagging this Topoi because an answer of Peter Arndt to a totally different MO question gives me a small amount of hope that some of the mysteries of Topos Theory could help me.
Best Answer
The lattice of ideals of $R$ is isomorphic to the lattice of regular quotients of $R$. Here, a regular quotient is an equivalence class of regular epimorphisms $R \to S$ in the category of rings (which are precisely the surjective ring homomorphisms). So this also serves as a categorical definition of an ideal. Also the ideals of $R$ are exactly the subobjects of $R$ in $\text{Mod}(R)$, but I think that you want to keep in the category of rings, right? Now, a prime ideal is characterized by the fact it is a proper ideal $\mathfrak{p}$ and for all ideals $\mathfrak{a}, \mathfrak{b}$ we have that $\mathfrak{a} * \mathfrak{b} \subseteq \mathfrak{p}$ implies $\mathfrak{a} \subseteq \mathfrak{p}$ or $\mathfrak{b} \subseteq \mathfrak{p}$. Unfortunately, there is no lattice-theoetic definition of the product of ideals, see this previous MO question, but there I give a definition which comes from ideas of Rosenberg's noncommutative algebraic geometry and uses subcategories of $\text{Mod}(R)$.
However, you can also give the following characterization within the category of rings: A prime ideal corresponds to a regular quotient $R \to S$ where $S$ is an integral domain. Now an integral domain is characterized by the property that it embeds into a field, where embeddings are given by monomorphisms (which actually coincide with the injective ring homomorphisms as in every algebraic category). A field $K$ is characterized by the following property: $K \neq 0$, where $0$ is the terminal object, and the lattice of ideals has exactly two elements, namely $0$ and $K$.
You may also consult the overlooked book "Categories of commutative algebras" by Yves Diers. Here the author developes all the basic commutative algebra for so-called Zariski categories, which are special locally presentable categories which resemble categories of commutative rings. In the chapter "Classical objects" we defines integral objects etc. in every Zariski category. Namely: First a simple object is defined to have exactly two congruences, and an integral object is defined to be one which embeds into a simple one.