One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, so that this description of commutative algebra is misleading. It feels to me like in most commutative algebra circles, the main approaches taken are geometric, rather than, say, multiplicative ideal theoretic. For example, in Eisenbud's commutative algebra book, less than one page is devoted to Dedekind domains, except for some scattered exercises.

If I were to make a Venn diagram, I would depict commutative algebra, algebraic geometry, and non-commutative algebra as all having nontrivial intersections with each other (rather than commutative algebra being subsumed by algebraic geometry).

My question is threefold: is there a good example of an important problem in commutative algebra that requires a geometric approach? is there a good example of an important problem in commutative algebra that rather requires methods from non-commutative algebra? And is there a good example of a problem in commutative algebra that requires both (e.g., that requires some results in "non-commutative geometry", a term that has multiple meanings)?

To answer a comment: I am not looking for a list of examples. I'm looking for a single important and good example of each, as I don't know any good examples of the three. I will accept an answer that provides three important examples, as requested. I just want to know if there are any important examples of all three at all.

ADDED: As I explained in the (now deleted) comments, I don't think this question should be or should have been moved to community wiki. If I ask for a proof of Theorem X, then there will likely not be a unique answer, and the OP can accept whichever proof that was offered as an answer that they believe was the best among them. The situation is the same here. I'm asking whether or not there exist important examples, which are not unique, but there are criteria for whether an example is an example or not, just as there are criteria for whether a proof is a proof or not. If there are no such examples, then I would like to know that. If there are such examples, then I would like to know that. It doesn't matter what the examples are, as long as there are indeed examples.

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