Maximal order

Remarks

Example: $M_n(\Z)$ is a maximal $\Z$-order in $M_n(\Q)$. Example: for an algebraic number field $F$, the ring of integers $\mathcal{O}_F$ is the unique maximal $\Z$-order in $F$. The reason is that every element of an order is integral over $\Z$ and so lies in $\mathcal{O}_F$. Non-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. Maximal 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 $\begin{pmatrix} 0 & 1/2 \\ 0 & 0 \end{pmatrix}$, which is not in $M_2(\Z)$. If $R$ is a Dedekind domain and $A$ is a separable algebra over $K$, every $R$-order in $A$ is contained in a maximal $R$-order.