In mathematics, an order in the sense of ring theory is a subring O {\displaystyle {\mathcal {O}}} of a ring A {\displaystyle A}, such that A {\displaystyle A} is a finite-dimensional algebra over the field Q {\displaystyle \mathbb {Q} } of rational numbers O {\displaystyle {\mathcal {O}}} spans A {\displaystyle A} over Q {\displaystyle \mathbb {Q} }, and O {\displaystyle {\mathcal {O}}} is a Z {\displaystyle \mathbb {Z} } -lattice in A {\displaystyle A}. Wikipedia