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

${displaystyle rMsubset IM}$

. 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[edit]

• 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[edit]

Let R be a ring. The Rees algebra

${displaystyle R[It]=oplus _{ngeq 0}I^{n}t^{n}}$

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

${displaystyle oplus _{ngeq 0}{overline {I^{n}}}t^{n}}$

. 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

${displaystyle ngeq 0}$

.

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

${displaystyle Isubset J}$

have the same integral closure if and only if they have the same multiplicity.^[1]

See also[edit]

References[edit]

• 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[edit]