nLab Bézout domain

Contents

Context

Arithmetic

Algebra

Contents

Definition

A Bézout domain is an integral domain that is also a Bézout ring.

In constructive mathematics

In constructive mathematics, there are different types of integral domains, yielding different types of Bézout domains: the gcd function and Bézout coefficient functions are no longer valued in R{0}R \setminus \{0\} in one variable, but in {xR|x0}\{x \in R \vert x \neq 0\}, {xR|x#0}\{x \in R \vert x \# 0\}, or some other definition, depending on what the base integral domain ends up being (classical, Heyting, discrete, residue, et cetera).

Properties

Every GCD domain of dimension at most 1 is a Bézout domain.

In classical mathematics

In classical mathematics, every Bézout domain RR that satisfies one of the following conditions is a principal ideal domain

  1. RR is a unique factorization domain

  2. RR is a Noetherian ring

  3. RR is an atomic domain?

  4. RR satisfies the ascending chain condition on principal ideals.

In constructive mathematics

In constructive mathematics, Bézout domains are usually better behaved because many important rings may fail to be principal ideal domains.

For instance, the ring of integers is a principal ideal domain if and only if the law of excluded middle holds: In one direction, the usual proofs rely on being able to decide whether any particular integer belongs to the ideal or not.

For the converse, let φ\varphi be an arbitrary proposition. Consider the ideal {x|(x=0)φ}\{ x \in \mathbb{Z} | (x = 0) \vee \varphi \}. By assumption, it is generated by some number nn. Since the integers are discrete, it holds that n=0n = 0 or n0n \neq 0. In the first case ¬φ\neg\varphi holds, in the second φ\varphi.

However, this ideal cannot be proved to be finitely generated either. If an ideal is generated by n 1,,n kn_1, \ldots, n_k, then we may form their gcd one step at a time, which we can do algorithmically. Therefore, \mathbb{Z} remains a Bézout domain.

As a result, in constructive mathematics not every Bézout domain that is a unique factorization domain is a principal ideal domain, as \mathbb{Z} is both a Bézout domain and a unique factorization domain, but is not a principal ideal domain.

Some authors have tried to define a principal ideal domain as a Noetherian Bézout domain, but it is unknown if this still coincides in constructive mathematics with the definition of principal ideal domain as a integral domain whose ideals are all principal ideals.

Examples

See also

References

See also:

Last revised on May 20, 2022 at 19:02:02. See the history of this page for a list of all contributions to it.