GCD Domain

In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor (GCD). Equivalently, any two non-zero elements of R have a least common multiple (LCM).

A GCD domain generalizes a unique factorization domain to the 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).