Commutative algebra | Ring theory

GCD domain

In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM). A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian). GCD domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields (Wikipedia).

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Square-free integer | Integral domain | Schreier domain | Cancellative semigroup | If and only if | Subclass (set theory) | Prüfer domain | Atomic domain | Entire function | Greatest common divisor | Principal ideal domain | Distributive lattice | Noetherian | Irreducible element | Monoid ring | Mathematics | Bézout domain | Prime element | Integrally closed domain | Least common multiple | Ascending chain condition on principal ideals | Complete lattice | Unique factorization domain | Abelian group | Principal ideal | Commutative ring