integral closure

Given a commutative unital ring $k$ and a field $L\supset k$, an element $x\in L$ 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 $M\subset L$ such that $x M \subset M$.

A ring $K\supset k$ 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 $f:K\to K'$ is a surjective homomorphism of rings and $K$ integral over $k\subset K$, 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 $S\subset k$, the localization $S^{-1} K$ is integral over $S^{-1} k$.

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.