Integral closure of an ideal explained

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: there exist a _{i} \in I ^{i} such thatr ^{n} + a _{1} r ^{n-1} + … + 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 \overline{I} if and only if there is a finitely generated R-module M, annihilated only by zero, such that r M \subset I M . 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

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 ^{di} _{d =- x} 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,

\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 ^{a _{1}} _{1} … X ^{a _{n}} _{n} . The integral closure of a monomial ideal is monomial.

Structure results

R[It] = ⊕ _{n} 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 ⊕ _{n} \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 an ideal generated by elements. Then

\overline{I ^{n+l} } \subset I^ for any n \ge 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, .

Further reading

• Irena Swanson, Rees valuations.