Bézout domain

In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain (UFD), but still is a GCD domain. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property. Bézout domains are named after the French mathematician Étienne Bézout. All PIDs are Bézout domains. Examples of Bézout domains that are not PIDs include the ring of entire functions (functions holomorphic on the whole complex plane) and the ring of all algebraic integers. In case of entire functions, the only irreducible elements are functions associated to a polynomial function of degree 1, so an element has a factorization only if it has finitely many zeroes. In the case of the algebraic integers there are no irreducible elements at all, since for any algebraic integer its square root (for instance) is also an algebraic integer. This shows in both cases that the ring is not a UFD, and so certainly not a PID. Valuation rings are Bézout domains. Any non-Noetherian valuation ring is an example of a non-noetherian Bézout domain. The following general construction produces a Bézout domain S that is not a UFD from any Bézout domain R that is not a field, for instance from a PID; the case R = Z is the basic example to have in mind. Let F be the field of fractions of R, and put S = R + XF[X], the subring of polynomials in F[X] with constant term in R. This ring is not Noetherian, since an element like X with zero constant term can be divided indefinitely by noninvertible elements of R, which are still noninvertible in S, and the ideal generated by all these quotients of is not finitely generated (and so X has no factorization in S).

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Prüfer domain

In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer. The ring of entire functions on the open complex plane form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring of integer polynomials is not .

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).

Bézout domain

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