Integral element

Builds on Commutative ring Subring Monic polynomial Polynomial ring

Definition

Let $B$ be a commutative ring and let $A$ be a subring of $B$ . An element $b \in B$ is integral over $A$ if there are an integer $n \ge 1$ and elements $a_0, \ldots, a_{n-1} \in A$ such that $b^n + a_{n-1} b^{n-1} + \cdots + a_1 b + a_0 = 0.$ Equivalently, $b$ is a root of a monic polynomial in $A[x]$ . More generally, for a ring homomorphism $f \colon A \to B$ , an element $b$ is integral over $A$ if it is integral over the subring $f(A)$ .

Remarks

Example: $i \in \C$ is integral over $\Z$ , because $i^2 + 1 = 0$ and $x^2 + 1$ is monic with integer coefficients.

Non-example: $1/2 \in \Q$ is not integral over $\Z$ . By the rational root theorem, a rational root of a monic polynomial with integer coefficients is an integer.

The following are equivalent for $b \in B$ :

$b$ is integral over $A$ .
The subring $A[b]$ is a finitely generated $A$ -module.
$A[b]$ is contained in a subring $C$ of $B$ that is a finitely generated $A$ -module.

The proof of 3 implies 1 uses the determinant trick. When $A$ and $B$ are fields, an element is integral exactly when it is algebraic, since a nonzero polynomial over a field can be divided by its leading coefficient.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Commutative ring concept
Monic polynomial concept
Polynomial ring concept
Subring concept
Integral element definition

Tag	`X83RX7KDSP`
URI	`m2://X83RX7KDSP`
Kind	definition
Writing	W0 stub
MSC	13B21
Defines	Integral element
Uses	Commutative ring, Subring, Monic polynomial, Polynomial ring
Changed	2 November 2026

Integral element

Definition

Remarks

Reading path

Further reading

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