# Lattice (module)

In mathematics, in the field of ring theory, a lattice is a module over a ring which 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 and R-torsion-free (no non-zero element of M is annihilated by a non-zero element of 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]

${\displaystyle N\mapsto W=K\cdot N;\quad W\mapsto N=W\cap M.\,}$