Abstract
We characterize the lattices which are isomorphic to submodule lattices of torsion free modules of Goldie dimension at least three over left Ore domains. The characterizing lattice-theoretic properties (axioms) are simple, natural, independent and of a geometric flavor.
In order to get uniqueness of the coordinatizing module (and ring) we start with a lattice L together with a distinguished subset P of “points” which shall correspond exactly to the non-zero cyclic submodules. In the proof of the coordinatization theorem we first construct a factor lattice L/~ to which we can apply a lattice theoretic version of the classical coordinatization theorem of projective geometry. Thus we get a skew field K and a K-vectorspace V such that L/~ is isomorphic to the lattice of linear subspaces of V. Then we construct a subring R ⊑ K, an R-submodule M ⊑ K and a lattice isomorphism f between L and the lattice of R-submodules of M with f[P] = {Rx|x∈M\{0}}.
The author wants to thank the Deutsche Forschungsgemeinschaft for their support during the formulation of the present paper.
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© 1985 D. Reidel Publishing Company
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Brehm, U. (1985). Coordinatization of Lattices. In: Kaya, R., Plaumann, P., Strambach, K. (eds) Rings and Geometry. NATO ASI Series, vol 160. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5460-1_11
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DOI: https://doi.org/10.1007/978-94-009-5460-1_11
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