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 .

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 byReiner (2003) p. 45
: $N \mapsto W = K \cdot N ; \quad W \mapsto N = W \cap M. \,$

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

* {{cite book , last=Reiner , first=I. , authorlink=Irving Reiner , title=Maximal Orders , series=London Mathematical Society Monographs. New Series , volume=28 , publisher=Oxford University Press , year=2003 , isbn=0-19-852673-3 , zbl=1024.16008

Module theory