Linearly compact chain rings

Abstract

A left [right] chain ring is a ring with identity in which the left [right] ideals are totally ordered by inclusion, and a chain ring is a ring that is both a left and right chain ring. Recently, Lorimer showed that the nondiscrete compact rings that are right (or left) chain rings are precisely the compact, discrete valuation rings, that is, the valuation rings of nondiscrete locally compact division rings. Here we show that the complete, discrete valuation rings of division rings are precisely the nondiscrete, strictly linearly compact left chain rings whose nonzero right ideals are open. We also show that the complete, discrete valuation rings finitely generated over their centers are precisely the centrally linearly compact, left chain rings whose center is not a field. 1991 Mathematics Subject Classification: 16W80.Recently, Lorimer [6] showed that the nondiscrete compact right chain rings (rings in which the set of right ideals is totally ordered by inclusion) are precisely the valuation rings of nondiscrete, locally compact division rings. His argument depended crucially on the fact that the quotient ring determined by an open ideal of a compact ring is finite.Here we shall Supplement his discussion by characterizing, in terms of linear compactness, the left chain rings that are valuation rings of complete, discrete valuations of division rings, and also those of division rings that are finitedimensional over their centers.All rings discussed here have identity elements, and all modules are unitary left modules. A linear topology on a module is a module topology for which the open submodules form a fundamental System of neighborhoods of zero. A linearly compact module is a Hausdorff, linearly topologized module such that every filter base of cosets of closed submodules has a nonempty intersection, and a strictly linearly compact module is a linearly compact module such that the quotient module of every open submodule is an artinian module, that is, satisfies the descending chain Brought to you by | The University of Auckland Library Authenticated Download Date | 6/15/15 11:28 PM