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
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.
can be used to compute the integral closure of an ideal.
The structure result is the following: the integral closure of
; 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 I an ideal generated by l elements.
A theorem of Rees states: let (R, m) be a noetherian local ring.
Assume it is formally equidimensional (i.e., the completion is equidimensional.).
Then two m-primary ideals
have the same integral closure if and only if they have the same multiplicity.