In algebra, the integral closure of an ideal of a commutative ring , denoted by , is the set of all elements r in that are integral over : that is, for each there exists such that
In other words, is a zero of a certain kind of monic polynomial. This integral closure is similar to the integral closure of a subring. For example, if is a domain, an element r in belongs to if and only if there is a finitely generated -module , annihilated only by zero, such that . It follows that is an ideal of (in fact, the integral closure of an ideal is always an ideal; see below). is said to be integrally closed if .
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.
Let be a ring. The Rees algebra can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of in , which is graded, is . In particular, is an ideal and ; 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 BrianconâÂÂSkoda theorem: let be a regular ring and an ideal generated by elements. Then for any .
A theorem of Rees states: let be a noetherian local ring. Assume it is formally equidimensional (i.e., the completion is equidimensional.). Then two m-primary ideals have the same integral closure if and only if they have the same multiplicity.