Bézout domain

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In mathematics, a Bézout domain, named after Étienne Bézout, is an integral domain in which every finitely generated ideal is principal.

A Bézout domain can also be defined as an integral domain in which any two elements have a greatest common divisor that is a linear combination of them (an integral domain in which Bézout's identity holds), and in particular Bézout domains are GCD domains. An example of a GCD domain that is not a Bézout domain is the ring of polynomials in two variables over a field.

Every Noetherian Bézout domain is a principal ideal domain. An example of a Bézout domain that is not a principal ideal domain is the ring of all polynomials (over a field) in all rational powers of a variable x.