In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.

If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., ${\displaystyle {\sqrt {2}}}$ or ${\displaystyle 1+i}$); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory.

In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.

Let ${\displaystyle B}$ be a ring and let ${\displaystyle A\subset B}$ be a subring of ${\displaystyle B.}$ An element ${\displaystyle b}$ of ${\displaystyle B}$ is said to be integral over ${\displaystyle A}$ if for some ${\displaystyle n\geq 1,}$ there exists ${\displaystyle a_{0},\ a_{1},\ \dots ,\ a_{n-1}}$ in ${\displaystyle A}$ such that $${\displaystyle b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0.}$$

The set of elements of ${\displaystyle B}$ that are integral over ${\displaystyle A}$ is called the integral closure of ${\displaystyle A}$ in ${\displaystyle B.}$ The integral closure of any subring ${\displaystyle A}$ in ${\displaystyle B}$ is, itself, a subring of ${\displaystyle B}$ and contains ${\displaystyle A.}$ If every element of ${\displaystyle B}$ is integral over ${\displaystyle A,}$ then we say that ${\displaystyle B}$ is integral over ${\displaystyle A}$, or equivalently ${\displaystyle B}$ is an integral extension of ${\displaystyle A.}$

There are many examples of integral closure which can be found in algebraic number theory since it is fundamental for defining the ring of integers for an algebraic field extension ${\displaystyle K/\mathbb {Q} }$ (or ${\displaystyle L/\mathbb {Q} _{p}}$).

Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.