Integral closure of an ideal

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

${\displaystyle r^{n}+a_{1}r^{n-1}+\cdots +a_{n-1}r+a_{n}=0.}$

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

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

Examples

• In ${\displaystyle \mathbb {C} [x,y]}$, ${\displaystyle x^{i}y^{d-i}}$ is integral over ${\displaystyle (x^{d},y^{d})}$. It satisfies the equation ${\displaystyle r^{d}+(-x^{di}y^{d(d-i)})=0}$where ${\displaystyle a_{d}=-x^{di}y^{d(d-i)}}$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, ${\displaystyle {\overline {xI}}=x{\overline {I}}}$. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
• Let ${\displaystyle R=k[X_{1},\ldots ,X_{n}]}$ be a polynomial ring over a field k. An ideal I in R is called monomial if it is generated by monomials; i.e., ${\displaystyle X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}}$. The integral closure of a monomial ideal is monomial.

Structure results

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

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 ${\displaystyle I\subset J}$ have the same integral closure if and only if they have the same multiplicity.[1]

Notes

1. Swanson 2006, Theorem 11.3.1 harvnb error: no target: CITEREFSwanson2006 (help)

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.