A Topological Characterization of Hjelmslev’s Classical Geometries

Lorimer, J. W.

doi:10.1007/978-94-009-5460-1_3

Cited by 2 publications

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References 37 publications

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“…This observation has significant geometric consequences for compact desarguesian Klingenberg planes. (See [32], [33], [35]). …”

Section: (Iii) An Open Discrete Valuation Ring (In the Sense Of Schilmentioning

confidence: 99%

The classification of compact right chain rings

Lorimer¹

1992

Forum Mathematicum

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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.

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“…This observation has significant geometric consequences for compact desarguesian Klingenberg planes. (See [32], [33], [35]). …”

Section: (Iii) An Open Discrete Valuation Ring (In the Sense Of Schilmentioning

confidence: 99%

The classification of compact right chain rings

Lorimer¹

1992

Forum Mathematicum

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“…In this paper we present a classification of compact punctally cohesive Desarguesian PK-planes and leave the determination of the locally compact (non-compact) planes for future publications. (The Pappian case has already been dealt with in [16].) Now recall that a PK-plane over a local ring is punctally cohesive exactly when its coordinate ring is a right chain ring (equivalently a local right Bezout ring).…”

Section: Introductionmentioning

confidence: 99%

The classification of compact punctally cohesive Desarguesian projective Klingenberg planes

Lorimer

1990

Geom Dedicata

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It is shown that for a compact Desarguesian projective Klingenberg plane ~ with incidence structure ~ = (P, D_, 1) and neighbour relation ~, where two distinct points always lie on some line, 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 fi~niteprojective Hjelmslev plane with ~ ¢id,-or .~ is a non-discrete totally disconnected ordinary projective plane with ~ ¢id.

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A Topological Characterization of Hjelmslev’s Classical Geometries

Cited by 2 publications

References 37 publications

The classification of compact right chain rings

The classification of compact right chain rings

The classification of compact punctally cohesive Desarguesian projective Klingenberg planes

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