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.