Ring of Algebraic Integers
Let $K / \Q$ be a number field.
Let $\Z \sqbrk x$ denote the polynomial ring in one variable over $\Z$.
That is, let $\mathbb A$ denote the algebraic integers over $K$.
Then $\mathbb A$ is a ring, called the Ring of Algebraic Integers.
This is a special case of Integral Closure is Subring.
The theorem says that $\mathbb A$ is a subring of $K$.