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 (mathematician)|Rees] that characterizes an analytically unramified ring.

Examples

In, is integral over. It satisfies the equation, where is in the th power of the ideal.
Radical ideals are integrally closed. The intersection of integrally closed ideals is integrally closed.
In a normal ring, for any non-zerodivisor x and any ideal I,. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
Let be a polynomial ring over a field k. An ideal I in R is called monomial if it is generated by monomials; i.e.,. The integral closure of a monomial ideal is monomial.

Structure results

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.