Let $\mathbb{A}$ denote the set of algebraic integers in $\mathbb{C}$. Then: 1. $\mathbb{A}$ is a subring of $\mathbb{C}$: sums and products of algebraic integers are algebraic integers. 2. If $a \in \mathbb{C}$ is both an algebraic integer and rational, then $a \in \mathbb{Z}$. 3. Any subring of $\mathbb{C}$ that is finitely generated as a $\mathbb{Z}$-module consists entirely of algebraic integers.