nLab
integral closure

Given a commutative unital ring k and a field Lk, an element xL is said to be integral over k if it satisfies a monic polynomial equation with coefficients in k, or equivalently, there exist a finitely-generated nonzero k-submodule ML such that xMM.

A ring Kk is said to be integral over k if every element of K is integral over k. The relation of integrality of overrings is transitive. If f:KK is a surjective homomorphism of rings and K integral over kK, then K=f(K) is integral over f(k).

The set of all elements of L integral over k is a subring of L called the integral closure of k in L. We say that k is integrally closed in L if it equals its own integral closure in L.

A commutative integral domain k is integrally closed if it is integrally closed in the quotient field of k.

If k is an integrally closed Noetherian domain and L a finite separable field extension of its quotient field Q(k) then the integral closure of k in L is finitely generated over k.

If k is a principal ideal ring and L a finite separable extension of degree n of its quotient field Q(k), then the integral closure of k in L is a free rank n-module over k.

If K is integral over a subring k then for any multiplicative set Sk, the localization S 1K is integral over S 1k.

Every unique factorization domain is integrally closed.

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Created on July 25, 2011 21:57:06 by Zoran Škoda (212.91.117.206)