Maximal ring topologies and the questions of completeness
by
V.I. Arnautov
Academy of Science of Moldova

Let R be a direct sum of the infinite collection { R \gamma| \gamma in \Gamma} of rings and M some collection of the ring topologies at R.

Theorem 1. If \tau is the maximal element in M, then (R, \tau) is a complete topological ring in every of the following cases:

1. M is a set of all such nondiskrete ring topologies \tau that the topological ring (R, \tau) has the basis of neigbourhoods of zero from ideals and for every \gamma in \Gamma the ring R_\gamma is a simple ring and R_\gamma² =/= 0 ;
2. M is a set of all such nondiskrete ring topologies \tau that the topological ring (R, \tau) is restricted and for every \gamma in \Gamma a multiplicative group of the ring R is simple and R_\gamma² =/= 0 ;
3. M is a set of all such nondiskrete ring topologies \tau, that the topological ring (R, \tau) is left restricted (is right restricted, has the basis of the neigbourhoods of zero from right ideals) and for every \gamma in \Gamma the ring R_\gamma is a body.

Theorem 2. Let M be a set of all nondiscrete ring topologies at R or a set of all such nondiskrete ring topologies at R that (R, \tau) has the basis of the neigbourhoods from subgroups (subrings); F be a finite field and for every \gamma in \Gamma the ring R_\gamma is isomorphic to F. If \Gamma is a countable set and the Continuum Hypothesis takes place, then there exist such ring topologies \tau and \tau' which are maximal at M, that (R, \tau) is a complete topological ring and (R, \tau') is an incomplete topological ring.

Date received: April 12, 1996

Copyright © 1996 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 # caae-03.