In abstract algebra, the concept of

**integral closure**is a generalization of the set of all algebraic integers. It is one of many closures in mathematics.

Let *S* be an integral domain with *R* a subring of *S*. An element *s* of *S* is said to be **integral** over *R* if *s* is a root of some monic polynomial with coefficients in *R*. ("Monic" means that the leading coefficient is 1, the identity element of *R*).

One can show that the set of all elements of *S* that are integral over *R* is a subring of *S* containing *R*; it is called the **integral closure** of *R* in *S*. If every element of *S* that is integral over *R* is in already in *R* then *R* is said to be **integrally closed** in *S*. (So, intuitively, "integrally closed" means that *R* is "already big enough" to contain all the elements that are integral over *R*). An equivalent definiton is that *R* is integrally closed in *S* iff the integral closure of *R* in *S* is equal to *R* (in general the integral closure is a superset of *R*). The terminology is justified by the fact that the integral closure of *R* in *S* is always integrally closed in *S*, and is in fact the smallest subring of *S* that contains *R* and is integrally closed in *S*.

In the special case where *S* is the fraction field of *R* and *R* is integrally closed in *S*, then *R* is said simply to be **integrally closed**.

For example, the integers **Z** are integrally closed (the fraction field of **Z** is **Q**, and the elements of **Q** that are integral over **Z** are just the elements of **Z** (!), hence the integral closure of **Z** in **Q** is **Z**). The integral closure of **Z** in the complex numbers **C** is the set of all algebraic integers.

See also algebraic closure; this is a special case of integral closure when *R* and *S* are fieldss.