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 {{nowrap|1=V = K · M}}.Reiner (2003) pp. 44, 108
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
:
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
{{reflist}}
- {{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 }}