Example 1

Let $K$ be a (commutative) field and $R$ a $K$-algebra; to say that an element $x \in R$ is integral over $K$ is equivalent to saying that $x$ is a root of a non-constant polynomial in the ring $K[X]$; generalizing the terminology introduced when $R$ is an extension of $K$ (Algebra, Chapter V, § 3, no. 3), the elements $x \in R$ which are integral over $K$ are also called the algebraic elements of $R$ over $K$.

(2) The elements of $\mathbf{Q}(i)$ which are integral over $\mathbf{Z}$ are the elements of the form $a + ib$ where $a \in \mathbf{Z}$ and $b \in \mathbf{Z}$ ("*Gaussian integers*"'); the elements of $\mathbf{Q}(\sqrt{5})$ which are integral over $\mathbf{Z}$ are the elements of the form $(a + b\sqrt{5})/2$ where $a$ and $b$ belong to $\mathbf{Z}$ and are both even or both odd (for these two examples see Exercise 1).

(3) The complex numbers integral over $\mathbf{Z}$ are also called algebraic integers.

Remarks (1) Let $A'$ be the subring of $R$ (contained in the centre of $R$) the image of $A$ under the ring homomorphism $A \to R$ which defines the $A$-algebra structure on $R$. Clearly it is equivalent to say that an element of $R$ is integral over $A$ or that it is integral over $A'$.