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  "body": "\n## Summary\n\nA maximal order is an [[order|1ZBX4TKVZ9]] in an algebra that is not contained in any larger order.\nIn an [[algebraic number field|MWBDFS2KJ5]] the ring of integers is the unique maximal order, so maximal orders generalize rings of integers to noncommutative algebras.\nIn noncommutative algebras maximal orders exist under mild hypotheses but are usually not unique.\nFor example, the conjugates of $M_2(\\Z)$ by elements of $GL_2(\\Q)$ are maximal orders in $M_2(\\Q)$.\nMaximal orders in [[quaternion algebras|ZWXW0T3EAC]] are central in the arithmetic of quaternions and of Shimura curves.\n\n## Explanations\n\n### L1 human\n\nAn order is a ring of \"integers\" inside an algebra, but an algebra can have many orders, some inside others.\nFor example $\\Z[\\sqrt{-3}]$ is an order in $\\Q(\\sqrt{-3})$, but it sits inside the bigger order $\\Z[(1+\\sqrt{-3})/2]$.\nA maximal order is one that cannot be enlarged.\nFor number fields there is exactly one, the ring of all algebraic integers of the field.\nFor matrix algebras and quaternion algebras there can be several, and studying how they relate is part of the theory.\n",
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