The classification of compact right chain rings

Abstract

By a (right) chain ring is meant an associative ring with l whose (right) ideals are totally ordered. We first prove that a compact right chain ring is equivalent to a compact local ring whose Jacobson radical is right principal. Then, we establish that a compact right chain rings is an invariant chain ring, and must be one of the following: a Galois field, a finite proper projective Hjelmslev ring or an open rank one (classical) discrete valuation ring of a nondiscrete locally compact totally disconnected skewfield. 1991 Mathematics Subject Classification: 16A80, 16A10, 51C05. §0. Introduction A right (left) chain ring is a ring with unit whose right (left) ideals form a totally ordered set (a chain) under set inclusion. A chain ring is both a right and a left chain ring. The classical examples are the finite rings of integers modulo a power of a prime and the classical valuation rings of skewfields in the sense of Schilling [42] or more generally in the sense of Mathiak [38]. Also äs mentioned in [4], they appear äs building blocks for localization of non-commutative Dedekind rings ([21]) or of FPF-rings ([19]) äs well äs valuation domains of ordered skewfields ([43]). Moreover, such rings are also the coordinate rings of Barbilian, Klingenberg and Hjelmslev planes where two points always lie on some line. (See [1], [46], [38], [30], [3l], [27]).Compact chain rings occur äs the (open) classical discrete valuation rings of rank one in (locally compact totally disconnected) skewfields ([42]) and more recently, covertly, in the work Jo-Ann Cohen [l 1], äs certain finite projective Hjelmslev rings.

The classification of compact punctally cohesive Desarguesian projective Klingenberg planes

“…Our classification is then accomplished by first proving that if the canonical image of a compact Desarguesian PK-plane is non-discrete, then the plane is an ordinary projective plane; and if the canonical image is discrete, then the plane is compact if and only if its coordinate local ring is compact. Our result is then a consequence of our classification of compact right chain rings in [18]. Precisely we prove that for a compact punctally cohesive Desarguesian PK-plane with incidence structure ~ and neighbour relation ~ exactly one of the following holds: ~ is a non-discrete connected or totally disconnected ordinary projective plane with ~ =id, ~ is a finite projective plane with ~ =id, ~ is a finite projective Hjelmslev plane with ~ ~ id or ~ is a non-discrete totally disconnected ordinary projective plane with ~ ~id.…”

“…In the latter case, R, = R/J" is compact and discrete and hence finite, and J(R.)= J/J" is nonzero. Thus, each R, is a finite proper PH-ring whose radical is nilpotent of degree n by [18, 2.4.1] and so by [18, 3.1] R, is a proper E-ring of rank n. Moreover, R = lim,_(R,) by [18,2.4.1 Cor.]. Hence, by [3], [23,2] (4)~(5): Suppose #(R) is infinite with ~ #id and (P, 0_, I) is a nondiscrete compact totally disconnected ordinary projective plane.…”

Linearly compact chain rings

“…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.…”

“…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.…”