integral closure

Given a commutative unital ring kk and a field LkL\supset k, an element xLx\in L is said to be integral over kk if it satisfies a monic polynomial equation with coefficients in kk, or equivalently, there exist a finitely-generated nonzero kk-submodule MLM\subset L such that xMMx M \subset M.

A ring KkK\supset k is said to be integral over kk if every element of KK is integral over kk. The relation of integrality of overrings is transitive. If f:KKf:K\to K' is a surjective homomorphism of rings and KK integral over kKk\subset K, then K=f(K)K' = f(K) is integral over f(k)f(k).

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

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

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

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

If KK is integral over a subring kk then for any multiplicative set SkS\subset k, the localization S 1KS^{-1} K is integral over S 1kS^{-1} k.

Every unique factorization domain is integrally closed.

  • Serge Lang, Algebraic number theory, GTM 110, Springer 1970, 2000
  • O. Zariski, Samuel, Commutative algebra
  • N. Bourbaki, Commutative algebra
  • Z. Borevich, I. Shafarevich, Number theory
  • E. Artin, J. Tate, Class field theory, 1967
  • A. Weil, Basic number theory

Created on July 25, 2011 at 21:57:06. See the history of this page for a list of all contributions to it.