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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 01:38, 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
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