[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/integral-closure-of-an-ideal\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/integral-closure-of-an-ideal\/","headline":"Integral closure of an ideal","name":"Integral closure of an ideal","description":"From Wikipedia, the free encyclopedia In algebra, the integral closure of an ideal I of a commutative ring R, denoted","datePublished":"2019-03-05","dateModified":"2019-03-05","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/43b8e66ae8bc148a0ef25e32292c223ee9ae24d5","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/43b8e66ae8bc148a0ef25e32292c223ee9ae24d5","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/integral-closure-of-an-ideal\/","about":["Wiki"],"wordCount":4064,"articleBody":"From Wikipedia, the free encyclopediaIn algebra, the integral closure of an ideal I of a commutative ring R, denoted by I\u00af{displaystyle {overline {I}}}, is the set of all elements r in R that are integral over I: there exist ai\u2208Ii{displaystyle a_{i}in I^{i}} such thatrn+a1rn\u22121+\u22ef+an\u22121r+an=0.{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 I\u00af{displaystyle {overline {I}}} if and only if there is a finitely generated R-module M, annihilated only by zero, such that rM\u2282IM{displaystyle rMsubset IM}. It follows that I\u00af{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 I=I\u00af{displaystyle I={overline {I}}}.The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.Table of ContentsExamples[edit]Structure results[edit]See also[edit]References[edit]Further reading[edit]Examples[edit]In C[x,y]{displaystyle mathbb {C} [x,y]}, xiyd\u2212i{displaystyle x^{i}y^{d-i}} is integral over (xd,yd){displaystyle (x^{d},y^{d})}. It satisfies the equation rd+(\u2212xdiyd(d\u2212i))=0{displaystyle r^{d}+(-x^{di}y^{d(d-i)})=0}, where ad=\u2212xdiyd(d\u2212i){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, xI\u00af=xI\u00af{displaystyle {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[X1,\u2026,Xn]{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., X1a1\u22efXnan{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 R[It]=\u2295n\u22650Intn{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 R[It]{displaystyle R[It]} in R[t]{displaystyle R[t]}, which is graded, is \u2295n\u22650In\u00aftn{displaystyle oplus _{ngeq 0}{overline {I^{n}}}t^{n}}. In particular, I\u00af{displaystyle {overline {I}}} is an ideal and I\u00af=I\u00af\u00af{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\u2013Skoda theorem: let R be a regular ring and I an ideal generated by l elements. Then In+l\u00af\u2282In+1{displaystyle {overline {I^{n+l}}}subset I^{n+1}} for any n\u22650{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 I\u2282J{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\u00a00-387-94268-8.Swanson, Irena; Huneke, Craig (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol.\u00a0336, Cambridge, UK: Cambridge University Press, ISBN\u00a0978-0-521-68860-4, MR\u00a02266432, Reference-idHS2006, archived from the original on 2019-11-15, retrieved 2013-07-12Further reading[edit]Wikimedia ErrorOur servers are currently under maintenance or experiencing a technical problem.Please try again in a few\u00a0minutes.See the error message at the bottom of this page for more\u00a0information. 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