In a basic algebra course last year, I learned some of how rings are taxonomized by properties such as "all the elements have unique factorization", or "all the ideals are principal ideals".
Studying ideals is still confusing for me because I want to visualize them as a poset structure (eg page (???) of Enumerative Combinatorics vol. 1 by Stanley), but I don't trust that my default visualization is general enough to meet all of the cases of ideal structures in CRing.
Even basic questions (like the corollaries on p. 417 of Lang, or the propositions on p. 5 of Atiyah MacDonald) leave me feeling tungtied... I can read these proofs for long enough that I can convince myself I know why they must be true, but whenever I see the problem again, I don't see right away how it can be proved.
Part of my question is for reading references that will help me feel more confident that I am not making any unspecified assumptions when I try to visualize ideal structure... spacially, graphically, topologically, or whatever.
For example: is a poset structure that comes from CRing by way inclusion ordering on a set of ideals always a connected planar graph? The answer seems to be, obviously yes; I'm looking for a text that treats this a a basic first example and then goes on from there.
Since the divisibility ordering of Z gives me my go-to visualization for the ideal structures that can come from CRing, there might be interesting (to me) examples that come from applying extremal graph theory (as Diestel puts it on p. 173 "how global parameters of a graph, such as its edge density or chromatic number, can influence its local substructures.") to synthesize elements of CRing.
The other part of my question is: Where might I find examples of such constructions? Or is a course in basic ring theory pretty much all about showing that we don't have to think about graphs like that when we are working in the category of rings?
Edit: Browsing around for similar reference requests I find this question:
From the comments, we get a survey article titled "The Many Lives of Lattice Theory" written by Rota and published in Notices of the AMS in 1997.
We intuitively feel that there is a geometry, projective, algebraic, or whatever, whose statements hold independently of the choice of a base field. Desargues’s theorem is the simplest theorem of such a “universal” geometry. A new class of commutative rings remains to be discovered that will be completely determined by their lattice of ideals.
Maybe the question should just be: "has this panned out at all?"