Integral closure of an ideal

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

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

In other words, ${\displaystyle r}$ 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 ${\displaystyle R}$ is a domain, an element r in ${\displaystyle R}$ belongs to ${\displaystyle {\overline {I}}}$ if and only if there is a finitely generated ${\displaystyle R}$-module ${\displaystyle M}$, annihilated only by zero, such that ${\displaystyle rM\subseteq IM}$. It follows that ${\displaystyle {\overline {I}}}$ is an ideal of ${\displaystyle R}$ (in fact, the integral closure of an ideal is always an ideal; see below). ${\displaystyle I}$ 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 ${\displaystyle d}$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 ${\displaystyle I}$, ${\displaystyle {\overline {xI}}=x{\overline {I}}}$. 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 ${\displaystyle I}$ in ${\displaystyle R}$ 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 ${\displaystyle R}$ 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 ${\displaystyle R}$ 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 ${\displaystyle (R,m)}$ 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.

See also

• Dedekind–Kummer theorem

Notes

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.