Given a commutative unital ring and a field , an element is said to be integral over if it satisfies a monic polynomial equation with coefficients in , or equivalently, there exist a finitely-generated nonzero -submodule such that .
A ring is said to be integral over if every element of is integral over . The relation of integrality of overrings is transitive. If is a surjective homomorphism of rings and integral over , then is integral over .
The set of all elements of integral over is a subring of called the integral closure of in . We say that is integrally closed in if it equals its own integral closure in .
A commutative integral domain is integrally closed if it is integrally closed in the quotient field of .
If is an integrally closed Noetherian domain and a finite separable field extension of its quotient field then the integral closure of in is finitely generated over .
If is a principal ideal ring and a finite separable extension of degree of its quotient field , then the integral closure of in is a free rank -module over .
If is integral over a subring then for any multiplicative set , the localization is integral over .
Every unique factorization domain is integrally closed.