In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the field of
ring theory, a lattice is a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
which is
embedded in a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, 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'' ยท ''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
[Reiner (2003) p. 45]
:
See also
*
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...
, 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
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
, year=2003 , isbn=0-19-852673-3 , zbl=1024.16008
Module theory