Integrally Closed

A commutative unit ring is integrally closed in a commutative unit ring containing as a subring if every element of that is integral over belongs to . In other words, there is no proper integral extension of contained in .

If is an integral domain, then is called an integrally closed domain if it is integrally closed in its field of fractions.

Every unique factorization domain is an integrally closed domain; e.g., the ring of integers and every polynomial ring over a field are integrally closed domains.

Being integrally closed is a local property. For an integral domain , the following are equivalent: is integrally closed, every localization at a prime ideal is integrally closed, and every localization at a maximal ideal is integrally closed (Stacks Project, Tag 037B). Also, is an integrally closed integral domain whenever is and is multiplicatively closed, contains 1, and does not contain 0.

For example, is not integrally closed in its field of fractions . The element satisfies the monic polynomial over but does not belong to that ring. Its integral closure in is , as in the algebraic normalization of the cusp .