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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:

Why are modular lattices important?

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.

Rota writes:

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?"

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  • $\begingroup$ Why do you think the graph is always planar? It seems very easy to get a $K_5$ in there, just from any chain of five ideals one contained in the next. $\endgroup$ – Kevin Carlson Jan 25 at 0:32
  • $\begingroup$ @KevinCarlson This kind of goes to my question. My default visualization of ideal structure is as a distributive lattice like the one that can be derived form the ring Z by looking at greatest common divisors. It's possible that this is not planar either... I just assume that it is based on my memory of diagrams I have seen. I'm asking for texts that deal explicitly with this type of graph theory question in the context of graphs that are defined in terms of objects from CRing (obvs. many ways to do this. We have two already.) $\endgroup$ – jessup Jan 25 at 2:12
  • $\begingroup$ I think the question of what the lattices of ideals of commutative rings looks like is intractable except that they are all modular lattices. It is also not a very useful tool when learning about commutative rings. You should just consider going out and gathering a large set of commutative examples and getting familiar with their ideals just as practice. $\endgroup$ – rschwieb Jan 25 at 3:06
  • $\begingroup$ @rschwieb Thank you for point out the terminology "modular lattice". I started looking in to that and edited the question accordingly. $\endgroup$ – jessup Jan 25 at 4:35
  • $\begingroup$ @user637706 There are tons of ring properties defined by conditions on their lattice of ideals. However, I don't think that means the ideal lattice "completely determines them." An easy example are uniserial rings (ones whose lattice is linearly ordered.) It's easy to construct two uniserial rings with isomorphic ideal lattices, and yet they are far from isomorphic. Even the Artinian and Noetherian conditions are poset conditions. Another one is the notion of a "distributive ring". $\endgroup$ – rschwieb Jan 25 at 14:23

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