Integral Element - Integral Closure

Integral Closure

Let be rings and A' the integral closure of A in B. (See above for the definition.)

Integral closures behave nicely under various constructions. Specifically, the localization S−1A' is the integral closure of S−1A in S−1B, and is the integral closure of in . If are subrings of rings, then the integral closure of in is where are the integral closures of in . If B is integral over A, then is integral over R for any commutative A-algebra R.

The integral closure of a local ring A in, say, B, need not be local. This is the case for example when A is Henselian and B is a field extension of the field of fractions of A.

If A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation rings of K containing A.

There is also a concept of the integral closure of an ideal. The integral closure of an ideal, usually denoted by, is the set of all elements such that there exists a monic polynomial with with as a root. The integral closure of an ideal is easily seen to be in the radical of this ideal.

There are alternate definitions as well.

• if there exists a not contained in any minimal prime, such that for all sufficiently large .

• if in the normalized blow-up of, the pull back of is contained in the inverse image of . The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings.

The notion of integral closure of an ideal is used in some proofs of the going-down theorem.