Background: I've been working my way through Thomas Jech's "Set Theory" because I'm working on some problems that have the potential to be logically independent of the usual axioms, or at least involve some hard set-theory about infinite sets, and I want to (eventually) better understand some of the set-theoretical techniques and methods that can be used to show such independence.
The first seven chapters of Jech's book have clear applications in my field of expertise, ring theory. I was already quite familiar with much of the material because of this fact.
Chapter 1: The basic axioms of set theory are used all the time in the language of ring theory, allowing for the formation of simple objects like unions, products, sequences, etc.
Chapter 2: The ordinal numbers have applications in infinitary constructions, and anywhere induction can be pushed further. For instance, one can define the "higher Wedderburn radicals" as a transfinite sequence of ideals which eventually stabilize at the prime radical.
Chapter 3: Everyone is familiar with how cardinal numbers give a very easy way to determine whether two objects are the same size, which helps prevent the existence of isomorphisms. Thus, this is a sort of first check one does to prove non-isomorphisms. There are other uses.
Chapter 4: Real numbers are a ring, with lots of interesting properties. What more needs to be said?
Chapter 5: The axiom of choice turns into Zorn's lemma. Using it you can prove all sorts of neat facts about existence of maximal ideals, algebraic closures, etc.
Chapter 6: Well-foundedness of relations comes up when trying to force processes to stop. One application I'm familiar with is in generalizing power series rings to allow for more products, but still requiring a "bottom" so that multiplication is well-defined.
Chapter 7: Boolean algebras, not much needs to be said here.
However, I'm just not familiar with any immediate uses of stationary sets in ring theory. This bothers me because (1) they seem quite useful in set theory, and (2) Jensen's Diamond, which settled the Whitehead problem (and was one reason I jumped into this current project) is stated in terms of stationary sets.
So, here is my formal question.
Question: What basic applications do stationary sets (in and of themselves, without appealing to axioms independent of ZFC) have in algebra? If applications to ring theory specifically could be provided, that would be even better.