Integral closure

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 which 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 is integral over R then R is said to be integrally closed in S. 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 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 fields.