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  "body": "\n## Definition\n\nLet $R$ be a Noetherian integral domain with field of fractions $K$, and let $A$ be a finite-dimensional algebra over $K$.\nAn $R$-[[order|1ZBX4TKVZ9]] $\\Lambda$ in $A$ is *maximal* if there is no $R$-order $\\Lambda'$ in $A$ with $\\Lambda \\subsetneq \\Lambda'$.\n\n## Remarks\n\nExample: $M_n(\\Z)$ is a maximal $\\Z$-order in $M_n(\\Q)$.\nExample: for an [[algebraic number field|MWBDFS2KJ5]] $F$, the ring of integers $\\mathcal{O}_F$ is the unique maximal $\\Z$-order in $F$.\nThe reason is that every element of an order is integral over $\\Z$ and so lies in $\\mathcal{O}_F$.\nNon-example: $\\Z[\\sqrt{-3}]$ is an order in $\\Q(\\sqrt{-3})$, but it is strictly contained in the order $\\Z[\\omega]$ with $\\omega = (1 + \\sqrt{-3})/2$, so it is not maximal.\nMaximal orders need not be unique: with $g = \\operatorname{diag}(1, 2)$, the order $g M_2(\\Z) g^{-1}$ is maximal in $M_2(\\Q)$ and contains the matrix $\\begin{pmatrix} 0 \u0026 1/2 \\\\ 0 \u0026 0 \\end{pmatrix}$, which is not in $M_2(\\Z)$.\nIf $R$ is a [[Dedekind domain|7NPY2CN5KB]] and $A$ is a [[separable algebra|X8H7A98Q55]] over $K$, every $R$-order in $A$ is contained in a maximal $R$-order.\n",
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