In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out...

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields...

noetherian local domain such that the integral closure is not finite over that domain.[citation needed] Let A be a noetherian integrally closed domain with field...

S {\displaystyle S} . An integral domain R {\displaystyle R} is said to be integrally closed if it is equal to its integral closure in its field of fractions...

factorization domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains...

Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all...

at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if...

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields...

domain. Euclidean domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains...

In abstract algebra, a Schreier domain, named after Otto Schreier, is an integrally closed domain where every nonzero element is primal; i.e., whenever...