Slope number: Difference between revisions
en>David Eppstein →Relation to degree: fix error per talk |
en>David Eppstein better citation for Jamison and Scott |
||
| Line 1: | Line 1: | ||
In mathematics, in the field of [[ring theory]], a '''lattice''' is a [[module (mathematics)|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. | |||
Let ''R'' be an [[integral domain]] with [[field of fractions]] ''K''. An ''R''-module ''M'' is a ''lattice'' in the ''K''-vector space ''V'' if ''M'' is finitely generated, ''R''-torsion-free (no non-zero element of ''R'' annihilates ''M'') and an ''R''-submodule of ''V''. It is ''full'' if ''V'' = ''K''·''M''.<ref>Reiner (2003) pp. 44, 108</ref> | |||
A submodule ''N'' of ''M'' which is again a lattice is an ''R''-pure sublattice if ''M''/''N'' is ''R''-torsionfree. There is a one-to-one correspondence between ''R''-pure sublattices ''N'' of ''M'' and ''K''-subspaces ''W'' of ''V'' given by<ref>Reiner (2003) p. 45</ref> | |||
:<math> N \mapsto W = K \cdot N ; W \mapsto N = W \cap M. \, </math> | |||
==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 }} | |||
[[Category:Module theory]] | |||
{{algebra-stub}} | |||
Revision as of 00:38, 3 April 2013
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.
Let R be an integral domain with field of fractions K. An R-module M is a lattice in the K-vector space V if M is finitely generated, R-torsion-free (no non-zero element of R annihilates M) and an R-submodule of V. It is full if V = K·M.[1]
A submodule N of M which is again a lattice is an R-pure sublattice if M/N is R-torsionfree. 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
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534