Lattice (Module)

In mathematics, in the field of ring theory, a lattice is a module over a ring that is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space.

Formal definition

Let R be an integral domain with field of fractions K. An R-submodule M of a K-vector space V is a lattice if M is finitely generated over R. It is full if V = K · M.^[1]

Pure sublattices

An R-submodule N of M that is itself a lattice is an R-pure sublattice if M/N is R-torsion-free. There is a one-to-one correspondence between R-pure sublattices N of M and K-subspaces W of V, given by^[2]

[math]\displaystyle{ N \mapsto W = K \cdot N ; \quad W \mapsto N = W \cap M. \, }[/math]

See also

• Lattice (group), for the case where M is a Z-module embedded in a vector space V over the field of real numbers R, and the Euclidean metric is used to describe the lattice structure

References

• Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. 28. Oxford University Press. ISBN 0-19-852673-3. 

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