I have a rather broad question: Are there any useful results/ approaches known on study commutative unitary rings and their ideals with *filter/ ultrafilter* methods?

What I know:

1. In Boolean algebra ideals are dual objects to filters (that's of course a very special ring)
2. If $R$ is the direct product of a family of fields $F\_{\alpha}$ indexed by a set $X$, ie $R= \prod\_{\alpha \in X} F\_{\alpha}$, then there exist a very strong relation: the proper ideals in $R$ are in bijective correpondence with the filters $F$ on $X$, see [this discussion.](https://math.stackexchange.com/questions/242231/ring-and-filters?noredirect=1&lq=1)

Is this the only interesting connection using filters as abstract toolbox to analyse structure of rings and their ideals or are there more?

More generally are there any general heuristics known when one may expect that the application of filter methods to some mathematical subarea could provide interesting results, and on the other hand when one should expect it's nearly useless (as seemingly here in the case for rings which are not fields). Can this 'dissonance' be explaned in heuristic terms?

My problem is that I have rather bad intuition for filters (...at latest when nonprincipal ultrafilters come into the game, my intuition tends instantly to say goodbye) and I'm wondering
if it's possible to develop some kind of intuition when filter methods might provide some new insight at the studied topic, and when tendentilly not. For study of rings it seems that filter methods not provide something new. Is there a heuristic reason why at least in this case for rings & ideals applying filter methods tends (...except for the case of products of fields) not to unravel some new interesting insights about the structure?

Hope, that the question is too vaguely formulated, it's primarily about intuition.