Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-08T22:02:43.729Z Has data issue: false hasContentIssue false
  • English
  • Français

Hjelmslev Planes Derived from Modular Lattices

Published online by Cambridge University Press:  20 November 2018

Benno Artmann
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In several papers, W. Klingenberg has elaborated the connections between Hjelmslev planes and a class of rings, called H-rings (4; 5; 6), which are rings of coordinates for the corresponding Hjelmslev planes. Certain homomorphic images of valuation rings are examples of H-rings. In these examples, the lattice of (right) ideals of the ring, say R,is a chain, and the coordinatization of the corresponding Hjelmslev plane yields a natural embedding of the plane in the lattice L(R3) of (right) submodules of the module R3. Now, L(R3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a1 = (1, 0, 0)R, a2 = (0, 1, 0)R, a2 = (0, 0, 1)R, and the sublattices L(N, ai) of elements less than or equal to ai are chains. Forgetting about the ring, we find ourselves in the situation of a problem suggested by Skornyakov (7, Problem 23, p. 166), namely, to study modular lattices with a homogeneous basis of chains. Baer (2) and Inaba (3) investigated lattices of this kind with Desarguesian properties and assuming that the chains L(N, ai) were finite. Representations of the lattices by means of certain rings can be found in both articles.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 76 - 83
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Artmann, B., On coordinates in modular lattices with a homogeneous basis (to appear in Illinois J. Math.).Google Scholar
2. Baer, R., A unified theory of projective spaces and finite abelian groups, Trans. Amer. Math. Soc. 52 (1942), 283–343.Google Scholar
3. Inaba, E., On primary lattices, J. Fac. Sci. Hokkaido Univ. Ser. I 11 (1948), 39–107.Google Scholar
4. Klingenberg, W., Projektive und affine Ebenen mit Nachbarelementen, Math. Z. 60 (1954), 384–406.Google Scholar
5. Klingenberg, W., Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg 20 (1955), 97–111.Google Scholar
6. Klingenberg, W., Projektive Geometrien mit Homomorphismus, Math. Ann. 182 (1956), 180–200.Google Scholar
7. Skornyakov, L. A., Complemented modular lattices and regular rings (Oliver and Boyd, London, 1964).Google Scholar