What are sources for pathological and non-so-pathological Gabriel filters on commutative rings? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:44:27Z http://mathoverflow.net/feeds/question/27115 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27115/what-are-sources-for-pathological-and-non-so-pathological-gabriel-filters-on-comm What are sources for pathological and non-so-pathological Gabriel filters on commutative rings? Carl Weisman 2010-06-04T22:52:16Z 2013-03-15T02:29:29Z <p>The heavy lifting in the theory of Gabriel filters is for noncommutative rings, and discussions I've been able to find all focus there.</p> <p>I am trying to develop a theory of Gabriel-filter localization for Nikolai Durov's generalized rings, and some ugly classical examples would help.</p> <p>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.</p> http://mathoverflow.net/questions/27115/what-are-sources-for-pathological-and-non-so-pathological-gabriel-filters-on-comm/123115#123115 Answer by Torsten Schoeneberg for What are sources for pathological and non-so-pathological Gabriel filters on commutative rings? Torsten Schoeneberg 2013-02-27T15:42:50Z 2013-03-15T02:29:29Z <p>(It might look as if I got paid for promoting this book, but:) Bo Stenström's <em>Rings of Quotients</em> (Springer Grundlehren vol. 217, 1975; see <a href="http://link.springer.com/book/10.1007/978-3-642-66066-5/page/1" rel="nofollow">SpringerLink here</a>; 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.</p>