GCD domain

In mathematics, a GCD domain (sometimes called just 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).^[1]

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 ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

^ Anderson, D. D. (2000). "GCD domains, Gauss' lemma, and contents of polynomials". In Chapman, Scott T.; Glaz, Sarah (eds.). Non-Noetherian Commutative Ring Theory. Mathematics and its Application. Vol. 520. Dordrecht: Kluwer Academic Publishers. pp. 1–31. doi:10.1007/978-1-4757-3180-4_1. MR 1858155.

[1] Anderson, D. D. (2000). "GCD domains, Gauss' lemma, and contents of polynomials". In Chapman, Scott T.; Glaz, Sarah (eds.). Non-Noetherian Commutative Ring Theory. Mathematics and its Application. Vol. 520. Dordrecht: Kluwer Academic Publishers. pp. 1–31. doi:10.1007/978-1-4757-3180-4_1. MR 1858155.

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