nLab
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 $xM\subset M$.

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 $f:K\to K\prime$ is a surjective homomorphism of rings and $K$ integral over $k\subset K$, then $K\prime =f(K)$ 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