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 (in fact, the integral closure of an ideal is always an ideal; see below.) I is said to be integrally closed if .
- In , is integral over .
- 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, . 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.
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 I an ideal generated by l elements. Then for any .
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.
- Swanson 2006, Theorem 11.3.1
- 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
- Irena Swanson, Rees valuations.