Integral closure of an ideal


In algebra, the integral closure of an ideal I of a commutative ring R, denoted by, is the set of all elements r in R that are integral over I: there exist such that
It is similar to the integral closure of a subring. For example, if R is a domain, an element r in R belongs to if and only if there is a finitely generated R-module M, annihilated only by zero, such that. It follows that is an ideal of R I is said to be integrally closed if.
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.

Examples

Let R be a ring. The Rees algebra can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of in, which is graded, is. In particular, is an ideal and ; i.e., the integral closure of an ideal is integrally closed. It also follows that the integral closure of a homogeneous ideal is homogeneous.
The following type of results is called the Briancon–Skoda theorem: let R be a regular ring and an ideal generated by elements. Then for any.
A theorem of Rees states: let be a noetherian local ring. Assume it is formally equidimensional. Then two m-primary ideals have the same integral closure if and only if they have the same multiplicity.