Bézout domain

In mathematics, a Bézout domain is an integral domain which is, in a certain sense, a non-Noetherian analogue of a principal ideal domain. More precisely, a Bézout domain is a domain in which every finitely generated ideal is principal. A Noetherian ring is a Bézout domain if and only if it is a principal ideal domain (PID). Bézout domains are named after the French mathematician Étienne Bézout.

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.^[1]

Properties

A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a greatest common divisor that is a linear combination of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal. The expression of the greatest common divisor of two elements of a PID as a linear combination is often called Bézout's identity, whence the terminology.

Note that the above gcd condition is stronger than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a GCD domain and thus Bézout domains are GCD domains. In particular, in a Bézout domain, irreducibles are prime.

For a Bézout domain R, the following conditions are all equivalent:

1. R is a principal ideal domain.
2. R is Noetherian.
3. R is a unique factorization domain (UFD).
4. R satisfies the ascending chain condition on principal ideals (ACCP).
5. Every nonzero nonunit in R factors into a product of irreducibles (R is an atomic domain).

The equivalence of (1) and (2) was noted above. Since a Bézout domain is a GCD domain, it follows immediately that (3), (4) and (5) are equivalent. Finally, if R is not Noetherian, then there exists an infinite ascending chain of finitely generated ideals, so in a Bézout domain an infinite ascending chain of principal ideals. (4) and (2) are thus equivalent.

A Bézout domain is a Prüfer domain, i.e., a domain in which each finitely generated ideal is invertible.

Roughly speaking, one may view the implications "Bézout domain implies Prüfer domain and GCD-domain" as the non-Noetherian analogues of the more familiar "PID implies Dedekind domain and UFD". The analogy fails to be precise in that a UFD (or an atomic Prüfer domain) need not be Noetherian.

Prüfer domains can be characterized as integral domains whose localizations at all prime (equivalently, all maximal) ideals are valuation domains. So the localization of a Bézout domain at a prime ideal is a valuation domain. Since an invertible ideal in a local ring is principal, a local ring is a Bézout domain iff it is a valuation domain. Moreover a valuation domain with noncyclic value group is not Noetherian, and every totally ordered abelian group is the value group of some valuation domain. This gives many examples of non-Noetherian Bézout domains. Here are two others:

• (Helmer, 1940) The ring of functions holomorphic on the entire complex plane.
• The ring of all algebraic integers. Theorem 102 of (Kaplansky, 1970) gives a more general result: let R be a Dedekind domain with quotient field K, let L be the algebraic closure of K, and let T be the integral closure of R in L. Suppose that for any finite extension of K, the ring of integers has a torsion class group. Then T is a Bézout domain. On the other hand a domain (not itself a field) whose fraction field is algebraically closed cannot be a PID, for then it would carry a nontrivial discrete valuation and hence admit ramified extensions of all degrees.

References

• O. Helmer, Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345-356.
• I. Kaplansky, Commutative Rings, Allyn & Bacon, 1970.
• P.M. Cohn, Bezout rings and their subrings, 1967.