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 $I$ : that is, for each $i$ there exists $a_{i} \in I^{i}$ such that

r^{n} + a_{1} r^{n - 1} + \dots + a_{n - 1} r + a_{n} = 0.

In other words, $r$ is a zero of a certain kind of monic polynomial. This integral closure 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 $r M \subseteq I M$ . 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 $ℂ [x, y]$ , $x^{i} y^{d - i}$ is integral over $(x^{d}, y^{d})$ . It satisfies the equation $r^{d} + (- x^{d i} y^{d (d - i)}) = 0$ , where $a_{d} = - x^{d i} y^{d (d - i)}$ is in the $d$ th power of 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 $I$ , $\overline{x I} = x \overline{I}$ . In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
Let $R = k [X_{1}, \dots, X_{n}]$ be a polynomial ring over a field k. An ideal $I$ in $R$ is called monomial if it is generated by monomials; i.e., $X_{1}^{a_{1}} \dots X_{n}^{a_{n}}$ . The integral closure of a monomial ideal is monomial.

Structure results

Let $R$ be a ring. The Rees algebra $R [I t] = \oplus_{n \geq 0} I^{n} t^{n}$ can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of $R [I t]$ 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.

Notes

↑ Swanson & Huneke 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.
Swanson, Irena; Huneke, Craig (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432, Reference-idHS2006, archived from the original on 2019-11-15, retrieved 2013-07-12

Integral closure of an ideal

Examples

Structure results

See also

Notes

References

Further reading