Integral closure of an ideal

In algebra, the integral closure of an ideal I of a commutative ring R, denoted by $$\overline{I}$$, is the set of all elements r in R that are integral over I: there exist $$a_i \in I^i$$ such that
 * $$r^n + a_1 r^{n-1} + \cdots + a_{n-1} r + a_n = 0.$$

It is similar to the integral closure of a subring. For example, if R is a domain, an element r in R belongs to $$\overline{I}$$ if and only if there is a finitely generated R-module M, annihilated only by zero, such that $$r M \subset I M$$. It follows that $$\overline{I}$$ 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 $$I = \overline{I}$$.

The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.

Examples

 * In $$\mathbb{C}[x, y]$$, $$x^i y^{d-i}$$ is integral over $$(x^d, y^d)$$. It satisfies the equation $$r^{d} + (-x^{di} y^{d(d-i)}) = 0$$, where $$a_d=-x^{di}y^{d(d-i)}$$is in the ideal.
 * Radical ideals (e.g., prime 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, $$\overline{xI} = x \overline{I}$$. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
 * Let $$R = k[X_1, \ldots, X_n]$$ be a polynomial ring over a field k. An ideal I in R is called monomial if it is generated by monomials; i.e., $$X_1^{a_1} \cdots X_n^{a_n}$$. The integral closure of a monomial ideal is monomial.

Structure results
Let R be a ring. The Rees algebra $$R[It] = \oplus_{n \ge 0} I^n t^n$$ can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of $$R[It]$$ in $$R[t]$$, which is graded, is $$\oplus_{n \ge 0} \overline{I^n} t^n$$. In particular, $$\overline{I}$$ is an ideal and $$\overline{I} = \overline{\overline{I}}$$; 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. Then $$\overline{I^{n+l}} \subset I^{n+1}$$ for any $$n \ge 0$$.

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 $$I \subset J$$ have the same integral closure if and only if they have the same multiplicity.