LINEARLY COMPACT CHAIN RINGS

A left [right] chain ring is a ring with identity in which the left [r ight] ideals are totally ordered by inclusion, and a chain ring is a r ing that is both a left and right chain ring. Recently. Lorimer showed that the nondiscrete compact rings that are right (or left) chain rin gs are precisely the compact, discrete valuation rings, that is. the v aluation 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 pre cisely the centrally linearly compact, left chain rings whose center i s not a field.