Geodesic
Hjelmslev Planes Derived from Modular Lattices
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 76-83

Voir la notice de l'article provenant de la source Cambridge University Press

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(R 3) of (right) submodules of the module R 3. Now, L(R 3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a 1 = (1, 0, 0)R, a 2 = (0, 1, 0)R, a 2 = (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. 
Hjelmslev Planes Derived from Modular Lattices. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 76-83. doi: 10.4153/CJM-1969-008-5
@misc{10_4153_CJM_1969_008_5,
     title = {Hjelmslev {Planes} {Derived} from {Modular} {Lattices}},
     journal = {Canadian journal of mathematics},
     pages = {76--83},
     year = {1969},
     volume = {21},
     number = {1},
     doi = {10.4153/CJM-1969-008-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-008-5/}
}
TY  - JOUR
TI  - Hjelmslev Planes Derived from Modular Lattices
JO  - Canadian journal of mathematics
PY  - 1969
SP  - 76
EP  - 83
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-008-5/
DO  - 10.4153/CJM-1969-008-5
ID  - 10_4153_CJM_1969_008_5
ER  - 
%0 Journal Article
%T Hjelmslev Planes Derived from Modular Lattices
%J Canadian journal of mathematics
%D 1969
%P 76-83
%V 21
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-008-5/
%R 10.4153/CJM-1969-008-5
%F 10_4153_CJM_1969_008_5

[1] 1. Artmann, B., On coordinates in modular lattices with a homogeneous basis (to appear in Illinois J. Math.). Google Scholar

[2] 2. Baer, R., A unified theory of projective spaces and finite abelian groups, Trans. Amer. Math. Soc. 52 (1942), 283–343. Google Scholar

[3] 3. Inaba, E., On primary lattices, J. Fac. Sci. Hokkaido Univ. Ser. I 11 (1948), 39–107. Google Scholar

[4] 4. Klingenberg, W., Projektive und affine Ebenen mit Nachbarelementen, Math. Z. 60 (1954), 384–406. Google Scholar

[5] 5. Klingenberg, W., Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg 20 (1955), 97–111. Google Scholar

[6] 6. Klingenberg, W., Projektive Geometrien mit Homomorphismus, Math. Ann. 182 (1956), 180–200. Google Scholar

[7] 7. Skornyakov, L. A., Complemented modular lattices and regular rings (Oliver and Boyd, London, 1964). Google Scholar

Cité par Sources :