Integral Closure Of An Ideal

In algebra, the integral closure of an ideal ''I'' of a commutative ring ''R'', denoted by $\overline$, is the set of all elements ''r'' in ''R'' that are integral over ''I'': there exist $a_i \in I^i$ such that
:$r^n + a_1 r^ + \cdots + a_ r + a_n = 0.$
It is similar to the integral closure of a subring. For example, if ''R'' is a domain, an element ''r'' in ''R'' belongs to $\overline$ 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 $\overline$ 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 $I = \overline$.

The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.

Examples

*In $\mathbb$, y/math>, $x^i y^$ is integral over $(x^d, y^d)$. It satisfies the equation $r^ + (-x^ y^) = 0$, where $a_d=-x^y^$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'', $\overline = x \overline$. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
*Let R = k _1, \ldots, X_n/math> be a polynomial ring over a field ''k''. An ideal ''I'' in ''R'' is called monomial if it is generated by monomials; i.e., $X_1^ \cdots X_n^$. The integral closure of a monomial ideal is monomial.

Structure results

Let ''R'' be a ring. The Rees algebra R t= \oplus_ 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 t/math> in R /math>, which is graded, is $\oplus_ \overline t^n$. In particular, $\overline$ is an ideal and $\overline = \overline$; 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 $\overline \subset I^$ 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

Notes

References

* Eisenbud, David, ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, .
* {{Citation , id=Reference-idHS2006 , last2=Huneke , first2=Craig , last1=Swanson , first1=Irena , author1-link=Irena Swanson , title=Integral closure of ideals, rings, and modules , url=http://people.reed.edu/~iswanson/book/index.html , publisher=Cambridge University Press , location=Cambridge, UK , series=London Mathematical Society Lecture Note Series , isbn=978-0-521-68860-4 , mr=2266432 , year=2006 , volume=336 , access-date=2013-07-12 , archive-date=2019-11-15 , archive-url=https://web.archive.org/web/20191115053353/http://people.reed.edu/~iswanson/book/index.html , url-status=dead

Further reading

* Irena Swanson
Rees valuations

Commutative algebra
Ring theory
Algebraic structures