Ring Theory and Pointfree Topology
by
Bernhard Banaschewski
McMaster University (Hamilton, Canada)

As is well known, many results in classical topology are actually consequences of corresponding facts in pointfree topology, that is, in the setting of frames, the lattices (introduced in the late 1950s in the Seminaire Ehresmann) which may be viewed as abstractly defined lattices of open sets of spaces. This talk will report on recent results of this kind, specifically in the context of commutative rings which are familiarly linked to topology by means of certain spaces associated with any ring, such as the prime spectrum, meaning: the space of prime ideals with its Zariski topology. It will be shown that the rôle of these spaces may be taken over by suitable frames, like the frame of radical ideals, with the added advantage that the classically necessary use of choice principles is thereby avoided. In particular, this will cover the pointfree characterizations of

(1) exchange rings in terms of their spectral properties,

(2) Gelfand and exchange rings by their sheaf representations,

(3) f-rings via sheaf representation by totally ordered rings, and

(4) certain f-rings as function rings.

Date received: February 26, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-31.