If A and B are commutative unit rings, and A is a subring of B, then A is called integrally closed in B if every element of B which is integral over A belongs to A; in other words, there is no proper integral extension of A contained in B. If A is an integral domain, then A 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 Z and every polynomial ring over a field are integrally closed domains. Being integrally closed is a local property, i.e., every localization of an integrally closed domain is again an integrally closed domain.

algebraically closed | integral closure