List of Structures from Rings to Fields

Introduction

This article lists the linear progression of commutative structures that the corpus builds one rung at a time, from a ring to a field: ring, commutative ring, reduced ring, integral domain, unique factorisation domain, principal ideal domain, Euclidean domain, field. Every entry points to the article that introduces the structure.

Each rung is recorded with the property it gains over the rung below and with the witness that shows the gain is strict: an object that satisfies every lower rung and fails the one above. The witnesses are objects of the corpus, not constructions of this article.

The article introduces nothing and proves nothing. It records examples and non-examples side by side, and it records the branch points at which the progression ceases to be linear.

The Ladder

Structure	The property it adds	Introduced in
ring	addition and multiplication with the distributive laws, unital with $1 \neq 0$	Rings
commutative ring	$ab = ba$ for all $a, b$	Commutative Rings
reduced ring	no nonzero nilpotent	Reduced Rings and the Nilradical
integral domain	no zero divisors, equivalently cancellation for nonzero elements	Integral Domains
GCD domain	every pair of elements has a greatest common divisor	GCD Domains
Bézout domain	every finitely generated ideal is principal	Bézout Domains
unique factorisation domain	every nonzero nonunit is a product of irreducibles, uniquely up to order and associates	Unique Factorisation Domains
principal ideal domain	every ideal is principal	Principal Ideal Domains
Euclidean domain	a Euclidean degree with division with remainder	Euclidean Domains
field	every nonzero element is a unit	Fields

The progression that the menu names runs Ring → Commutative ring → Reduced ring → Integral domain → UFD → PID → Euclidean domain → Field. The two entries GCD domain and Bézout domain are the refinement of the step from an integral domain to a UFD, and they are treated separately below because at that step the progression branches rather than continuing in a line.

The Strictness of Each Step

Implication	The hypothesis whose loss the witness shows	Witness	Introduced in
a commutative ring is a ring	commutativity	$M_n(\mathbb{R})$, $n \geq 2$	Matrix Algebras
a reduced ring is a commutative ring	no nonzero nilpotent	$\mathbb{Z}/4\mathbb{Z}$, in which $2^2 = 0$	Reduced Rings and the Nilradical
an integral domain is a reduced ring	no zero divisors	$\mathbb{Z} \times \mathbb{Z}$, with $(1,0)(0,1) = 0$	Examples of Rings and Fields
a unique factorisation domain is an integral domain	unique factorisation	$\mathbb{Z}[\sqrt{-5}]$, in which $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$	Unique Factorisation Domains
a principal ideal domain is a unique factorisation domain	every ideal principal	$\mathbb{Z}[x]$, whose ideal $(2, x)$ is not principal	Principal Ideal Domains
a Euclidean domain is a principal ideal domain	a Euclidean degree	$\mathbb{Z}[(1+\sqrt{-19})/2]$	Euclidean Domains
a field is a Euclidean domain	every nonzero element a unit	$\mathbb{Z}$, in which $2$ is not a unit	The Integers

The first witness, $M_2(\mathbb{R})$, is the matrix ring of Matrix Algebras; the progression is not entered by $M_2(\mathbb{R})$ again. The witness $\mathbb{Z}[\sqrt{-5}]$ is the classical Noetherian domain whose class group has order $2$, recorded in Dedekind Domains and Ideal Class Groups as well as in Unique Factorisation Domains. The witness $\mathbb{Z}[(1+\sqrt{-19})/2]$ is the principal ideal domain that fails to be Euclidean, recorded in Euclidean Domains; the subring $\mathbb{Z}[\sqrt{-19}]$ is not integrally closed.

Examples on Each Rung

The table records where the standard objects of the corpus sit. An object appears at the strongest rung it reaches among the eight; where an object reaches a rung only through a branch, the branch is named.

Object	Strongest rung reached	Introduced in
$\mathbb{Z}$	Euclidean domain, not a field	The Integers
$\mathbb{Z}[i]$	Euclidean domain, not a field	Examples of Rings and Fields
$k[x]$	Euclidean domain, not a field	Polynomial Rings and Rational Functions
$k[x_1, \dots, x_n]$, $n \geq 2$	unique factorisation domain, not principal	Polynomial Rings and Rational Functions
$\mathbb{Z}[x]$	unique factorisation domain, not principal	Polynomial Rings and Rational Functions
$k[x, y]$	unique factorisation domain, not principal	Polynomial Rings and Rational Functions
$\mathbb{Z}[\sqrt{2}]$	Euclidean domain, not a field	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{-5}]$	integral domain, not a unique factorisation domain	Examples of Rings and Fields
$\mathcal{O}_K$	Dedekind domain, not principal in general; a unique factorisation domain exactly when the class number is $1$	Algebraic Number Theory, Dedekind Domains and Ideal Class Groups
$\mathbb{Z} \times \mathbb{Z}$	reduced ring, not an integral domain	Examples of Rings and Fields
$\mathbb{Z}/4\mathbb{Z}$	commutative ring, not reduced	Reduced Rings and the Nilradical
$\mathbb{Z}/6\mathbb{Z}$	reduced ring, not an integral domain: $2 \cdot 3 = 0$	Modular Arithmetic and the Ring of Residues
$M_2(\mathbb{R})$	ring, not commutative	Matrix Algebras
$\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$	field	The Rational Numbers, The Real Numbers, The Complex Numbers
$\mathbb{F}_p$, $\mathbb{F}_q$	field	Finite Fields

Every field sits at the top rung, so $\mathbb{F}_p$, $\mathbb{F}_q$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are Euclidean domains as well as fields; the ring $\mathbb{Z}[\sqrt{2}]$ is Euclidean, with norm $\lvert a^2 - 2b^2 \rvert$, and reaches the same rung as $\mathbb{Z}[i]$.

The Two Strengthenings of a GCD Domain

The step from an integral domain to a unique factorisation domain passes through a rung whose two strengthenings are incomparable. A GCD domain is an integral domain in which every pair of elements has a greatest common divisor; a Bézout domain is an integral domain in which every finitely generated ideal is principal.

Structure	Relation to a GCD domain	Witness of strictness	Introduced in
GCD domain	the rung itself	—	GCD Domains
Bézout domain	a GCD domain in which every finitely generated ideal is principal	$\mathbb{Z}[x]$ is a GCD domain (indeed a UFD) that is not Bézout	Bézout Domains
unique factorisation domain	an incomparable strengthening of a GCD domain	$\overline{\mathbb{Z}}$ is Bézout and not a UFD	Unique Factorisation Domains
principal ideal domain	the intersection: a UFD that is Bézout, equivalently a Noetherian Bézout domain	$\mathbb{Z}[x]$ is a GCD domain (a UFD) that is not principal	Principal Ideal Domains

Thus a principal ideal domain is exactly a unique factorisation domain that is also Bézout, and also exactly a Bézout domain that is Noetherian. The ring $\overline{\mathbb{Z}}$ of all algebraic integers is the standard witness that a Bézout domain need not be Noetherian and need not be a unique factorisation domain; it is introduced in Bézout Domains.

The Fraction Field of Each Domain

Every rung from an integral domain upward embeds in its field of fractions, which is introduced in Localization and the Fraction Field. The table records the fraction field of the standard domains of the corpus; the field is the top rung of the ladder, so each domain is separated from the top by the loss of some divisibility property and not by the absence of inverses.

Domain	Field of fractions	Introduced in
$\mathbb{Z}$	$\mathbb{Q}$	The Rational Numbers
$\mathbb{Z}[i]$	$\mathbb{Q}(i)$	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{-5}]$	$\mathbb{Q}(\sqrt{-5})$	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{2}]$	$\mathbb{Q}(\sqrt{2})$	Examples of Rings and Fields
$k[x]$, $k[x_1,\dots,x_n]$	$k(x)$, $k(x_1,\dots,x_n)$	Polynomial Rings and Rational Functions
$\mathbb{Z}[x]$	$\mathbb{Q}(x)$	Polynomial Rings and Rational Functions
$\mathcal{O}_K$	the number field $K$	Algebraic Number Theory
$\overline{\mathbb{Z}}$	$\overline{\mathbb{Q}}$	Algebraically Closed Fields
an arbitrary domain $R$	$\operatorname{Frac}(R)$	Localization and the Fraction Field

The passage to the fraction field does not repair a failure of unique factorisation: $\operatorname{Frac}(\mathbb{Z}[\sqrt{-5}]) = \mathbb{Q}(\sqrt{-5})$ is a field, hence a unique factorisation domain, while $\mathbb{Z}[\sqrt{-5}]$ is not, because the failure lives in the integral elements.

Warnings: Structures Expected and Absent

Object	Why it is not on this ladder	Introduced in
$\mathbb{H}$	a division ring, but not commutative, so it never enters the commutative progression	Quaternion Algebra
the free algebra $R\langle x_1, \dots, x_n\rangle$	a non-commutative domain; the non-commutative ladder is the subject of List of Structures from Rings to Division Rings	Tensor Powers and the Free Algebra
$\mathbb{O}$	a division algebra whose multiplication is not associative; not even a ring	Octonion Algebra
the zero ring $\{0\}$	excluded by the convention $1 \neq 0$	Rings, §2

A non-commutative division ring is not stopped by a missing zero divisor: it is stopped by commutativity, and the two chains meet only at the fields. This is recorded again, from the non-commutative side, in List of Structures from Rings to Division Rings.

Summary

This article has listed the eight rungs from a ring to a field, the property gained at each, and the witness that makes each gain strict. It has recorded the two further strengthenings of a GCD domain, the unique factorisation domains and the Bézout domains, which are incomparable to each other and meet at the principal ideal domains, and it has placed the standard objects of the corpus on the rung they reach. The witnesses $M_2(\mathbb{R})$, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z} \times \mathbb{Z}$, $\mathbb{Z}[\sqrt{-5}]$, $\mathbb{Z}[x]$, $\mathbb{Z}[(1+\sqrt{-19})/2]$ and $\mathbb{Z}$ separate the seven steps in order.

Summary of Notation

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

Symbol	Meaning
$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$	The number systems
$\mathbb{Z}/n\mathbb{Z}$, $\mathbb{F}_p$, $\mathbb{F}_q$	Residue ring, prime field, finite field
$M_2(\mathbb{R})$, $M_n(R)$	Matrix rings
$\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{2}]$, $\mathbb{Z}[\sqrt{-5}]$	Quadratic integer rings
$\mathbb{Z}[(1+\sqrt{-19})/2]$	Ring of integers of $\mathbb{Q}(\sqrt{-19})$, a principal ideal domain that is not Euclidean
$\overline{\mathbb{Z}}$	Ring of all algebraic integers
$\mathcal{O}_K$	Ring of integers of a number field
$k[x]$, $k[x_1,\dots,x_n]$, $k[x,y]$	Polynomial rings
$R\langle x_1,\dots,x_n\rangle$	Free algebra
$\mathbb{H}$, $\mathbb{O}$	Quaternions, octonions