In algebra, the integral closure of an ideal I of a commutative ring R, denoted by I ¯ , is the set of all elements r in R that are integral over I: there exist a i ∈ I i such that
r 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 I ¯ if and only if there is a finitely generated R-module M, annihilated only by zero, such that r M ⊂ I M . It follows that 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 ¯ .
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.
In C [ x , y ] , x i y d − i is integral over ( x d , y d ) .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, x I ¯ = x I ¯ . In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.Let R = k [ X 1 , … , 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 1 a 1 ⋯ X n a n . The integral closure of a monomial ideal is monomial.Let R be a ring. The Rees algebra R [ I t ] = ⊕ n ≥ 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 [ I t ] in R [ t ] , which is graded, is ⊕ n ≥ 0 I n ¯ t n . In particular, I ¯ is an ideal and I ¯ = 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 I n + l ¯ ⊂ I n + 1 for any n ≥ 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 ⊂ J have the same integral closure if and only if they have the same multiplicity.
