# 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]

## 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 (2003) pp. 44, 108
- Reiner (2003) p. 45

- Reiner, I. (2003).
*Maximal Orders*. London Mathematical Society Monographs. New Series.**28**. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.

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