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  "body": "\n## Summary\n\nAn order is a [[subring|0GNCRPYXYV]] of a finite-dimensional algebra over a field.\nIt is a finitely generated module over a ring of integers such as $\\Z$, and it spans the algebra over the field.\nThe basic examples are the subrings of an [[algebraic number field|MWBDFS2KJ5]] that are free of full rank over $\\Z$, such as $\\Z[\\sqrt{-3}]$ in $\\Q(\\sqrt{-3})$.\nOrders in noncommutative algebras include $M_n(\\Z)$ in $M_n(\\Q)$ and the integral group ring $\\Z G$ in $\\Q G$.\nThey are central in integral representation theory and in the arithmetic of quaternion algebras.\nThe largest orders, the [[maximal orders|E5DH1CSW1N]], generalize rings of integers.\n\n## Explanations\n\n### L1 human\n\nThe integers sit inside the rationals as a grid of points that still spans everything once fractions are allowed.\nAn order is the same idea for a bigger algebra: a subring that is a grid over $\\Z$ and whose rational multiples fill the whole algebra.\nThe Gaussian integers $\\Z[i]$ form an order in $\\Q(i)$, with grid basis $1, i$.\nThe integer matrices form an order in the rational matrices.\nOrders let one do number theory, with divisibility and ideals, inside algebras that are not fields.\n",
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