List of Principal Ideal and Euclidean Domains

Introduction

This article lists the principal ideal domains, the Euclidean domains and the Bézout domains of the corpus, the division defined on each Euclidean domain, and the strict inclusions between the three classes. Every entry points to the article that introduces the object.

The three classes are related by two inclusions and one identity: every Euclidean domain is a principal ideal domain, every principal ideal domain is a Bézout domain, and a Bézout domain that is Noetherian is exactly a principal ideal domain. The Euclidean domains carry a division with remainder, the principal ideal domains are the domains in which every ideal is generated by one element, and the Bézout domains are the domains in which every finitely generated ideal is principal.

The article introduces nothing and proves nothing. It records examples and non-examples side by side, and it records the object that separates each pair of classes.

The Three Classes and the Inclusions

Class	Defining property	The article that introduces it
Euclidean domain	a Euclidean degree $N$ with division with remainder, $a = qb + r$ with $r = 0$ or $N(r) < N(b)$	Euclidean Domains
principal ideal domain	every ideal is principal	Principal Ideal Domains
Bézout domain	every finitely generated ideal is principal; Bézout's identity for two elements	Bézout Domains
GCD domain	every pair of elements has a greatest common divisor	GCD Domains

Inclusion	Strict?	The witness	Introduced in
Euclidean $\subseteq$ principal	strict	$\mathbb{Z}[(1+\sqrt{-19})/2]$, principal and not Euclidean	Euclidean Domains
principal $\subseteq$ Bézout	strict	$\overline{\mathbb{Z}}$, Bézout, not Noetherian, not principal	Bézout Domains
Bézout $\subseteq$ GCD	strict	$\mathbb{Z}[x]$, a GCD domain (indeed a UFD) that is not Bézout	GCD Domains
principal $=$ Noetherian Bézout	an identity of classes	—	Bézout Domains

The Euclidean algorithm and the extended form that produces the Bézout coefficients are the subject of Euclidean Domains; the vocabulary of the greatest common divisor is that of GCD Domains. The identity of the last row is the reason the class of principal ideal domains can be described from either side, as a Noetherian Bézout domain or as a unique factorisation domain that is Bézout.

The Euclidean Domains and Their Divisions

Euclidean domain	The Euclidean degree $N$	The division	Introduced in
$\mathbb{Z}$	$N(a) = \lvert a \rvert$	division with remainder of integers	The Integers
$\mathbb{Z}[i]$	$N(a+bi) = a^2 + b^2$	division in the square lattice $\mathbb{Z}[i]$	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{-2}]$	$N(a+b\sqrt{-2}) = a^2 + 2b^2$	division in the lattice	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{2}]$	$N(a+b\sqrt{2}) = \lvert a^2 - 2b^2 \rvert$	division in the lattice	Examples of Rings and Fields
$\mathbb{Z}[\tfrac{1+\sqrt{-3}}{2}]$	$N(a+b\tfrac{1+\sqrt{-3}}{2}) = a^2 + ab + b^2$	division in the triangular lattice	Examples of Rings and Fields
$k[x]$	$N(f) = \deg f$	division with remainder of polynomials	Polynomial Rings and Rational Functions
$\mathbb{F}_q[x]$	$N(f) = \deg f$	division with remainder over the finite field	Finite Fields
$k[t]_{(t)}$, $k[[x]]$, $\mathbb{Z}_{(p)}$, $\mathbb{Z}_p$	$N = v$, the valuation of the discrete valuation ring	every ideal is a power of the maximal ideal	Examples of Rings and Fields, Localization and the Fraction Field, Absolute Values, Valuations and Completions
every field	$N(x) = 1$ for $x \neq 0$	trivial: every remainder is $0$	Fields

The Gaussian integers are the standard worked example: the ring is Euclidean for the norm $a^2 + b^2$, so it is a principal ideal domain and a unique factorisation domain, and the division in the lattice produces the remainder of smaller norm. The rings $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{-2}]$, $\mathbb{Z}[\sqrt{2}]$ and the Eisenstein integers are the norm-Euclidean quadratic rings of the corpus, each with the norm recorded in the table.

The Principal Ideal Domains

Principal ideal domain	Why it is principal	Euclidean?	Introduced in
$\mathbb{Z}$	the division algorithm on $\mathbb{Z}$	yes, $N = \lvert \cdot \rvert$	The Integers
$\mathbb{Z}[i]$	the norm $a^2 + b^2$	yes	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{-2}]$	the norm $a^2 + 2b^2$	yes	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{2}]$	the norm $\lvert a^2 - 2b^2 \rvert$	yes	Examples of Rings and Fields
$\mathbb{Z}[\tfrac{1+\sqrt{-3}}{2}]$	the norm $a^2 + ab + b^2$	yes	Examples of Rings and Fields
$k[x]$, $\mathbb{F}_q[x]$	the degree	yes	Polynomial Rings and Rational Functions, Finite Fields
$k[t]_{(t)}$, $k[[x]]$	the valuation, since every nonzero ideal is $(t^n)$	yes, $N = v$	Absolute Values, Valuations and Completions, Examples of Rings and Fields
$\mathbb{Z}_{(p)}$, $\mathbb{Z}_p$	the valuation, since every nonzero ideal is $(p^n)$	yes, $N = v$	Absolute Values, Valuations and Completions
$\mathcal{O}_K$ with class number $1$	the trivial class group	only for the norm-Euclidean cases above	Dedekind Domains and Ideal Class Groups
$\mathbb{Z}[(1+\sqrt{-19})/2]$	the trivial class number of $\mathbb{Q}(\sqrt{-19})$	no	Euclidean Domains
a field, for instance $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$	the only ideals are $(0)$ and $(1)$	yes, $N = 1$	Fields

The principal ideal domains that are not Euclidean are the reason the first inclusion above is strict, and $\mathbb{Z}[(1+\sqrt{-19})/2]$ is the standard example, recorded in Euclidean Domains. The discrete valuation rings are the local examples of principal ideal domains, and they are Euclidean with the valuation as Euclidean degree: their ideals form the single chain $\mathcal{O} \supsetneq \mathrm{M} \supsetneq \mathrm{M}^2 \supsetneq \cdots$, so every ideal is principal, and in each step of the division the remainder is either zero or of smaller value.

The Bézout Domains

Bézout domain	Why it is Bézout	Noetherian?	Introduced in
every principal ideal domain	finitely generated ideals are principal by the definition	yes	Principal Ideal Domains
$\overline{\mathbb{Z}}$	the ring of all algebraic integers is Bézout	no	Bézout Domains
the ring of integers of a number field with class number $1$	principal, hence Bézout	yes	Dedekind Domains and Ideal Class Groups

The ring $\overline{\mathbb{Z}}$ is the standard witness that a Bézout domain need not be Noetherian and need not be a unique factorisation domain; a Bézout domain that is a unique factorisation domain is a principal ideal domain, and this is the reason the classes meet as they do.

The Strictness of the Inclusions

Boundary	The class below	The class above	The witness	Introduced in
Euclidean versus principal	Euclidean domain	principal ideal domain	$\mathbb{Z}[(1+\sqrt{-19})/2]$	Euclidean Domains
principal versus Bézout	principal ideal domain	Bézout domain	$\overline{\mathbb{Z}}$	Bézout Domains
Bézout versus GCD	Bézout domain	GCD domain	$\mathbb{Z}[x]$	GCD Domains
principal versus unique factorisation	principal ideal domain	unique factorisation domain	$\mathbb{Z}[x]$	Principal Ideal Domains

A unique factorisation domain and a Bézout domain are incomparable strengthenings of the GCD domains, and their intersection is exactly the principal ideal domains. This is the same boundary that List of Maximal Structures Before Failure records from the other direction.

Warnings

Object	Why it is not in these classes	Introduced in
$\mathbb{Z}[\sqrt{-5}]$	not principal: the prime $P = (2, 1+\sqrt{-5})$ is not principal	Dedekind Domains and Ideal Class Groups
$\mathbb{Z}[x]$	a unique factorisation domain, but the ideal $(2, x)$ is not principal	Polynomial Rings and Rational Functions
$k[x_1, \dots, x_n]$, $n \geq 2$	not principal: $(x_1, x_2)$	Polynomial Rings and Rational Functions
$\mathcal{O}_K$ with nontrivial class group	not principal, not Bézout	Dedekind Domains and Ideal Class Groups
$M_2(\mathbb{R})$	not commutative: the Euclidean degree is defined on an integral domain	Matrix Algebras
$\mathbb{H}$	a division ring, not a commutative domain	Quaternion Algebra

Summary

This article has listed the Euclidean, principal ideal and Bézout domains of the corpus with the division defined on each Euclidean domain. The Euclidean domains are $\mathbb{Z}$, the Gaussian, $\sqrt{-2}$, $\sqrt{2}$ and Eisenstein integer rings, the polynomial rings $k[x]$ and $\mathbb{F}_q[x]$, the discrete valuation rings, and the fields; the principal ideal domain that is not Euclidean is $\mathbb{Z}[(1+\sqrt{-19})/2]$; the Bézout domains are the principal ideal domains together with the non-Noetherian $\overline{\mathbb{Z}}$; and the strict inclusions are witnessed by $\mathbb{Z}[(1+\sqrt{-19})/2]$, $\overline{\mathbb{Z}}$ and $\mathbb{Z}[x]$.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.

Symbol	Meaning
$N$	The Euclidean degree of a Euclidean domain
$a = qb + r$	Division with remainder
$\mathbb{Z}$, $\mathbb{Q}$	The integers, the rationals
$\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{2}]$, $\mathbb{Z}[\sqrt{-2}]$, $\mathbb{Z}[\tfrac{1+\sqrt{-3}}{2}]$	Norm-Euclidean quadratic and Eisenstein rings
$\mathbb{Z}[(1+\sqrt{-19})/2]$	Principal ideal domain that is not Euclidean
$\overline{\mathbb{Z}}$, $\mathcal{O}_K$	All algebraic integers, ring of integers
$k[x]$, $\mathbb{F}_q[x]$, $R[x]$	Polynomial rings
$k[t]_{(t)}$, $k[[x]]$, $\mathbb{Z}_{(p)}$, $\mathbb{Z}_p$	Localisations, power series, $p$-adic integers
$\mathrm{M}$	The maximal ideal of a local ring