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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.
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-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}}

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