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.
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Formal definition
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 M is annihilated by a regular element of R) and an R-submodule of V. It is full if V = K·M.
Pure sublattices
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
References
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