Integral closure of an ideal

In algebra, the integral closure of an ideal I of a commutative ring R, denoted by [math]\displaystyle{ \overline{I} }[/math], is the set of all elements r in R that are integral over I: there exist [math]\displaystyle{ a_i \in I^i }[/math] such that

[math]\displaystyle{ r^n + a_1 r^{n-1} + \cdots + a_{n-1} r + a_n = 0. }[/math]

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

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

Examples

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

Structure results

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

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 [math]\displaystyle{ I \subset J }[/math] have the same integral closure if and only if they have the same multiplicity.

See also

• Dedekind–Kummer theorem

References

• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN:0-387-94268-8.
• Swanson, Irena; Huneke, Craig (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge, UK: Cambridge University Press, Reference-idHS2006, ISBN 978-0-521-68860-4, http://people.reed.edu/~iswanson/book/index.html, retrieved 2013-07-12 

Further reading

• Irena Swanson, Rees valuations.