Inalgebra, theintegral closure of an idealI of a commutative ringR, denoted by${\displaystyle \overline{I}}$, is the set of all elementsr inR that are integral overI: 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 theintegral closure of a subring. For example, ifR is a domain, an elementr inR belongs to${\displaystyle \overline{I}}$ if and only if there is a finitely generatedR-moduleM, annihilated only by zero, such that${\displaystyle rM\subset IM}$. It follows that${\displaystyle \overline{I}}$ is an ideal ofR (in fact, the integral closure of an ideal is always an ideal; see below.)I is said to beintegrally closed if${\displaystyle I=\overline{I}}$.

The integral closure of an ideal appears in a theorem ofRees that characterizes ananalytically unramified ring.

• 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 anormal ring, for any non-zerodivisorx and any idealI,${\displaystyle \overline{xI}=x\overline{I}}$. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
• Let${\displaystyle R=k[X_1,\dots ,X_n]}$ be a polynomial ring over a fieldk. An idealI inR is calledmonomial 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.

LetR be a ring. TheRees algebra${\displaystyle R[It]={\oplus }_{n\ge 0}{I}^n{t}^n}$ 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${\displaystyle {\oplus }_{n\ge 0}\overline{I^n}{t}^n}$. 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 theBriancon–Skoda theorem: letR be a regular ring andI an ideal generated byl elements. Then${\displaystyle \overline{I^{n+l}}\subset I^{n+1}}$ for any${\displaystyle n\ge 0}$.

A theorem of Rees states: let (R,m) be a noetherian local ring. Assume it isformally equidimensional (i.e., the completion is equidimensional.). Then twom-primary ideals${\displaystyle I\subset J}$ have the same integral closure if and only if they have the samemultiplicity.^[1]

• Dedekind–Kummer theorem

• 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 fromthe original on 2019-11-15, retrieved2013-07-12

• Irena Swanson,Rees valuationsArchived 2013-04-18 at theWayback Machine.

• Commutative algebra
• Ring theory
• Algebraic structures

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