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 $(x^{d},y^{d})$.
• 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, ${\displaystyle {\overline {xI}}=x{\overline {I}}}$. 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

1. 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

Further reading

• Irena Swanson, Rees valuations.