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.

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
 * $$N \mapsto W = K \cdot N ; \quad W \mapsto N = W \cap M. \,$$