Question

The heavy lifting in the theory of Gabriel filters is for noncommutative rings, and discussions I've been able to find all focus there.

I am trying to develop a theory of Gabriel-filter localization for Nikolai Durov's generalized rings, and some ugly classical examples would help.

Classical expositions seem to leap as quickly as possible to the associated torsion theories. I doubt, after trying for a while, that torsion theories can work for modules over a GR: Factor modules may be very badly behaved, even when the GR is "naive," with all operations faithfully represented on the monoid of unary operations.

Answer 1

(It might look as if I got paid for promoting this book, but:) Bo Stenström's Rings of Quotients (Springer Grundlehren vol. 217, 1975; see SpringerLink here; an earlier draft had appeared as LNM 237) contains lots of examples of Gabriel filters (called Gabriel topologies there) on commutative as well as non-commutative rings. Of particular interest might be chapters VI -- IX and XIII, but all the chapters have many examples and exercises.