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

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

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 $rM\subseteq IM$. 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 $(x^{d},y^{d})$. It satisfies the equation $r^{d}+(-x^{di}y^{d(d-i)})=0$, where $a_{d}=-x^{di}y^{d(d-i)}$ 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 $I$, ${\overline {xI}}=x{\overline {I}}$. 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.

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, 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

Further reading

• Irena Swanson, Rees valuations Archived 2013-04-18 at the Wayback Machine.