
Lattice (module)

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.
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"·"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