Lattice (module) explained

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 .^[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]

N \mapsto W = K ⋅ N ; W \mapsto N = W \cap M.

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

• Book: Reiner, I. . Irving Reiner

. Irving Reiner . Maximal Orders . London Mathematical Society Monographs. New Series . 28 . . 2003 . 0-19-852673-3 . 1024.16008 .

Notes and References

1. Reiner (2003) pp. 44, 108
2. Reiner (2003) p. 45