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 $rM\subset IM$ . 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 $R[It]=\oplus _{n\geq 0}I^{n}t^{n}$ 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 $\oplus _{n\geq 0}{\overline {I^{n}}}t^{n}$ . 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 $n\geq 0$ .

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.[1]

Notes

Swanson 2006, Theorem 11.3.1

References

Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432

Integral closure of an ideal

Examples

Structure results

Notes

References

Further reading