Integral Closure Of An Ideal
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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 In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' ...
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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/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 ideal In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is call ...
s (e.g., prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed. *In a
normal ring In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ...
, 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 In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
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 In commutative algebra, the Rees algebra of an ideal ''I'' in a commutative ring ''R'' is defined to be R t\bigoplus_^ I^n t^n\subseteq R The extended Rees algebra of ''I'' (which some authors refer to as the Rees algebra of ''I'') is defined asR ...
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 Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
, 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 Irena Swanson is an American mathematician specializing in commutative algebra. She is head of the Purdue University Department of Mathematics since 2020. She was a professor of mathematics at Reed College from 2005 to 2020. Education and caree ...

Rees valuations
Commutative algebra Ring theory Algebraic structures