Integral closure of an ideal

In algebra, the integral closure of an ideal "I" of a commutative ring "R", denoted by formula_1, is the set of all elements "r" in "R" that are integral over "I": there exist formula_2 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 formula_1 if and only if there is a finitely generated "R"-module "M", annihilated only by zero, such that formula_5. It follows that formula_1 is an ideal of "R" (in fact, the integral closure of an ideal is always an ideal; see below.) "I" is said to be integrally closed if formula_7.
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.
Structure results.
Let "R" be a ring. The Rees algebra formula_16 can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of formula_17 in formula_18, which is graded, is formula_19. In particular, formula_1 is an ideal and formula_21; 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 formula_22 for any formula_23.
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 formula_24 have the same integral closure if and only if they have the same multiplicity.