maximal order

《Algebraic Number Theory》 · 第 72 页 · Chapter I，Section 12「Orders」，Definition 12.1

Let $K\mid\mathbb{Q}$ be an algebraic number field of degree $n$. An order of $K$ is a subring $\mathcal{O}$ of $\mathcal{O}_K$ which contains an integral basis of length $n$. The ring $\mathcal{O}_K$ is called the maximal order of $K$.

《Arakelov Geometry》 · 第 64 页 · Chapter 3，Section 3.1「Orders」

For an algebraic number field $K$, we set $O_K=\{x\in K\mid x\text{ is integral over }\mathbb{Z}\}$, which is called the maximal order in $K$ or the ring of integers of $K$.