List of Rings

Introduction

This article lists the rings that the corpus meets, commutative and non-commutative, with the data that separate them: their units, their zero divisors, their ideals and their chain conditions. Every entry points to the article that introduces the object.

The convention is that of Rings: a ring is associative and unital and satisfies $1 \neq 0$, so the zero ring is the single exception and is named where it occurs. The classes into which these rings fall — commutative ring, reduced ring, integral domain, unique factorisation domain, principal ideal domain, Euclidean domain, field and division ring — are gathered in List of Structures from Rings to Fields and List of Structures from Rings to Division Rings. This article gathers the objects, not the rungs.

The article introduces nothing and proves nothing. It records examples and non-examples side by side, and it records the objects that a reader might expect to find here and will not.

The Number Systems

Ring	Units	Zero divisors	Ideals and chain conditions	Introduced in
$\mathbb{Z}$	$\pm 1$	none	every ideal principal; Noetherian, not Artinian	The Integers
$\mathbb{Q}$	$\mathbb{Q}\setminus\{0\}$	none	$0,\ \mathbb{Q}$; Noetherian and Artinian	The Rational Numbers
$\mathbb{R}$	$\mathbb{R}\setminus\{0\}$	none	$0,\ \mathbb{R}$; Noetherian and Artinian	The Real Numbers
$\mathbb{C}$	$\mathbb{C}\setminus\{0\}$	none	$0,\ \mathbb{C}$; Noetherian and Artinian	The Complex Numbers
$\mathbb{D}$	$N(u) \neq 0$	the null cone $\{N = 0\}$	$0,\ \mathbb{D}e_{\pm},\ \mathbb{D}$; Artinian	Split-Complex Algebra
$\mathbb{D}'$	$x \neq 0$	the maximal ideal $\mathrm{M} = (\varepsilon)$	the powers $(\varepsilon^m)$; local, Artinian	Dual-Numbers Algebra
$\mathbb{H}$	$\mathbb{H}\setminus\{0\}$	none	only $0$ and $\mathbb{H}$; simple	Quaternion Algebra
$\mathbb{H}_{\mathbb{D}}$	$N(u) \in \mathbb{D}^{\times}$	yes	two maximal ideals	Split-Biquaternion Algebra
$\mathbb{B}$	$N(\tilde{Q}) \neq 0$	yes	only $0$ and $\mathbb{B}$; simple, not division	Biquaternion Algebra

The quaternions $\mathbb{H}$ are the non-commutative ring whose every nonzero element is a unit; the biquaternions $\mathbb{B}$ and the split-biquaternions $\mathbb{H}_{\mathbb{D}}$ are the two eight-dimensional relatives that acquire zero divisors, and the split-complex numbers $\mathbb{D}$ and the dual numbers $\mathbb{D}'$ are the two-dimensional commutative rings with a degenerate or indefinite norm.

Residue Rings, Finite Rings and Products

Ring	Units	Zero divisors	Ideals and chain conditions	Introduced in
$\mathbb{Z}/n\mathbb{Z}$	the classes coprime to $n$, of order $\varphi(n)$	the nonunits of a composite modulus	ideals correspond to the divisors of $n$; finite, Artinian	Modular Arithmetic and the Ring of Residues
$\mathbb{F}_p$	$\mathbb{F}_p^{\times}$, cyclic of order $p-1$	none	only $0$ and $\mathbb{F}_p$; field	Finite Fields
$\mathbb{F}_q$, $q = p^n$	cyclic of order $q-1$	none	only $0$ and $\mathbb{F}_q$; field	Finite Fields
$\mathbb{F}_2^n$	the elements with all coordinates $1$	yes for $n \geq 2$	a product of copies of $\mathbb{F}_2$	Examples of Rings and Fields
$R \times S$	the pairs $(u, v)$ of units	$(1, 0)(0, 1) = 0$	a product of the ideals of the factors	Examples of Rings and Fields
$\mathbb{Z}/4\mathbb{Z}$	$\{1, 3\}$	$2$ is nilpotent, $2^2 = 0$, so the ring is not reduced	local with maximal ideal $(2)$; Artinian	Reduced Rings and the Nilradical
$\mathbb{Z}/6\mathbb{Z}$	$\{1, 5\}$	$2 \cdot 3 = 0$, neither factor zero	the product $\mathbb{F}_2 \times \mathbb{F}_3$; finite, Artinian	Modular Arithmetic and the Ring of Residues
the zero ring $\{0\}$	$1 = 0$	—	excluded by the convention $1 \neq 0$	Rings, §2

The non-example that fixes this section is $\mathbb{Z}/6\mathbb{Z}$: it is reduced, being the product $\mathbb{F}_2 \times \mathbb{F}_3$, and it is not an integral domain, since $2 \cdot 3 = 0$ with both factors nonzero.

Characteristic

The characteristic is that of Rings, §1: the least $n \geq 1$ with $n \cdot 1 = 0$, or $0$ if there is none. The table records where each value occurs among the rings above.

Characteristic	Rings	Introduced in
$0$	$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{D}$, $\mathbb{D}'$, $\mathbb{H}$, $\mathbb{H}_{\mathbb{D}}$, $\mathbb{B}$	Rings, §1; The Integers, The Rational Numbers, The Real Numbers, The Complex Numbers, Split-Complex Algebra, Dual-Numbers Algebra, Quaternion Algebra, Split-Biquaternion Algebra, Biquaternion Algebra
$0$	$R[x]$, $R[[x]]$, $k(x)$, $k((t))$, $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{2}]$, $\mathbb{Z}[\sqrt{-5}]$, $\mathcal{O}_K$, $\overline{\mathbb{Z}}$	Polynomial Rings and Rational Functions, Formal Power Series and Completion, Fields, §19, Absolute Values, Valuations and Completions, Examples of Rings and Fields, Algebraic Number Theory, Bézout Domains
$p$	$\mathbb{F}_p$, $\mathbb{F}_{p^n}$	Finite Fields
$p$	$\mathbb{F}_p[[t]]$	Local Fields
$n$	$\mathbb{Z}/n\mathbb{Z}$	Modular Arithmetic and the Ring of Residues
$2$	$\mathbb{F}_2^n$	Examples of Rings and Fields

A finite ring has nonzero characteristic, and a ring of characteristic $0$ contains a copy of $\mathbb{Z}$; a ring of characteristic $n$ contains a copy of $\mathbb{Z}/n\mathbb{Z}$. The zero ring is the only ring in which the characteristic would be $1$, and it is excluded by the convention $1 \neq 0$.

Polynomial, Power Series and Function Rings

Ring	Units	Zero divisors	Ideals and chain conditions	Introduced in
$R[x]$	the units of $R$	none when $R$ is a domain	principal when $R$ is a field; Noetherian when $R$ is	Polynomial Rings and Rational Functions
$R[x_1, \dots, x_n]$	the units of $R$	none when $R$ is a domain	finitely generated ideals, Noetherian when $R$ is	Polynomial Rings and Rational Functions
$R[[x]]$	the series with unit constant term	none when $R$ is a domain	local when $R$ is a field	Formal Power Series and Completion
$k(x)$	every nonzero rational function	none	only $0$ and $k(x)$; field	Fields, §19
$k((t))$	the Laurent series with nonzero leading term	none	the valuation ring $k[[t]]$; field	Absolute Values, Valuations and Completions
$k[[x]]$	the series with nonzero constant term	none when $k$ is a field	local with maximal ideal $(x)$	Examples of Rings and Fields
$k[x, x^{-1}]$	the monomials $c x^m$, $c \neq 0$	none	principal; the group ring $k[\mathbb{Z}]$	Group Algebras
$\mathbb{Z}_p$	the units of $\mathbb{Z}_p$, $\lvert x \rvert_p = 1$	none	local with maximal ideal $(p)$	Absolute Values, Valuations and Completions
$\mathbb{Z}_{(p)}$	the fractions with numerator prime to $p$	none	local with maximal ideal $p\mathbb{Z}_{(p)}$	Localization and the Fraction Field
$\mathbb{Z}[i]$	$\pm 1, \pm i$	none	Euclidean, hence every ideal principal	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{2}]$	$\pm(1+\sqrt{2})^m$	none	Euclidean, hence every ideal principal	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{-2}]$	$\pm 1$	none	Euclidean	Examples of Rings and Fields
$\mathbb{Z}[\sqrt{-5}]$	$\pm 1$	none	Noetherian, not principal	Examples of Rings and Fields
$\mathbb{Z}[\zeta_n]$	finite rank, by Dirichlet's unit theorem	none	Dedekind, not principal in general	Cyclotomic Fields
$\mathcal{O}_K$	by Dirichlet's unit theorem	none	Dedekind, so nonzero ideals factor uniquely	Algebraic Number Theory
$\overline{\mathbb{Z}}$	the roots of unity it contains	none	not Noetherian, but Bézout	Bézout Domains

The list separates the rings in which factorisation is governed by a Euclidean algorithm ($\mathbb{Z}[i]$) from those in which it is governed only by ideals ($\mathbb{Z}[\sqrt{-5}]$ and $\mathcal{O}_K$) and from those that are not Noetherian at all ($\overline{\mathbb{Z}}$, the ring of all algebraic integers). The failure of unique factorisation in $\mathbb{Z}[\sqrt{-5}]$ is recorded in Unique Factorisation Domains and in Examples of Rings and Fields.

Matrix, Group and Non-Commutative Rings

Ring	Units	Zero divisors	Ideals and chain conditions	Introduced in
$M_n(R)$, $n \geq 2$	the matrices whose determinant is a unit of $R$	yes, already for $n = 2$	only $0$ and $M_n(k)$ when $k$ is a field; simple	Matrix Algebras
$M_n(D)$, $D$ a division ring	the invertible matrices $\mathrm{GL}_n(D)$	yes	only $0$ and $M_n(D)$; simple	Matrix Algebras
$k[G]$, $G$ finite	contains $k^\times G$	yes as soon as $\lvert G \rvert \geq 2$	governed by the representations of $G$	Group Algebras
$k[G]$, $G$ infinite	contains $k^\times G$	yes unless $k[G]$ is a domain	not Artinian, Noetherian only for special $G$	Group Algebras
$k[G]$ for $G$ torsion-free and $k$ a domain	contains $k^\times G$	none	—	Non-Commutative Domains
$A_1(k) = k\langle x, y\rangle/(yx - xy - 1)$	the nonzero constants $k^{\times}$	none	a domain; the ideal structure is one-sided	Quotients of the Tensor Algebra
$R\langle x_1, \dots, x_n\rangle$	the units of $R$	none	a domain when $R$ is; the one-sided ideals differ	Tensor Powers and the Free Algebra
$T(V)$, the tensor algebra	the units of $R$ when $V$ is free of rank $\geq 1$	none when $R$ is a domain and $V$ is free	$\mathbb{Z}$-graded, with the universal property of the free algebra	Tensor Powers and the Free Algebra
$\Lambda(M)$, the exterior algebra	—	—	associative and unital, graded-commutative, generated by $M$	The Exterior Algebra
$\mathbb{C}\ell(V, q)$, the Clifford algebra	—	—	associative and unital, parity-graded, with $v^2 = q(v)$	Clifford Algebras
$\operatorname{End}_A(M)$, the endomorphism ring	the automorphisms $\operatorname{Aut}_A(M)$	—	generally non-commutative; $M_n(R) = \operatorname{End}_R(R^n)$	Automorphisms of Modules over an Algebra

The matrix ring $M_n(\mathbb{R})$, $n \geq 2$, is the standard ring that is neither commutative nor free of zero divisors; it appears again in List of Non-Commutative Rings and List of Zero Divisors and Nilpotents. The Weyl algebra $A_1(k)$ is the standard non-commutative domain whose ideal theory is asymmetric, and the free algebra $R\langle x_1, \dots, x_n\rangle$ is the free object from which the others are obtained by imposing relations.

Rings Introduced in Other Parts of the Corpus

The corpus meets rings outside its algebra articles as well; the property the corpus records for each, and the article that introduces it, are given here.

Ring	What the corpus records	Introduced in
the Boolean ring $B$	every element satisfies $x^2 = x$; commutative of characteristic $2$, reduced and regular	Von Neumann Regular Rings
the power set ring of a set	symmetric difference and intersection; the standard Boolean ring	Von Neumann Regular Rings
the cohomology ring $H^*(X; R)$	associative and unital, and graded-commutative: $uv = (-1)^{\lvert u \rvert \lvert v \rvert} vu$	Cup and Cap Products
the representation ring $R_k(G)$	the free abelian group on the isomorphism classes of finite-dimensional $k[G]$-modules, with $[V][W] = [V \otimes_k W]$ and unit the trivial module	Character Theory
the Grothendieck–Witt ring $GW(F)$	the Grothendieck group of the isometry classes of non-degenerate quadratic forms under $\perp$, made a ring by the tensor product of forms	The Witt Group and the Grothendieck–Witt Ring
the Witt ring $W(F)$	the quotient $GW(F)/\mathbb{Z}[H]$ by the hyperbolic plane, with the fundamental ideal $I = \ker \dim$	The Witt Group and the Grothendieck–Witt Ring
the ring of symmetric functions $\Lambda$	the polynomial ring $K[e_1, e_2, \dots]$ in countably many variables	Symmetric Functions and Schur Functions
the graded ring of modular forms $M_*(\Gamma(1))$	$\mathbb{C}[E_4, E_6]$, the two generators algebraically independent of weights $4$ and $6$	Modular Forms

Warnings

An object that a reader may expect among the rings, and does not find, is recorded here with the reason.

Object	Why it is not listed as a ring	Introduced in
the octonions $\mathbb{O}$	a division algebra, but multiplication is not associative, so $\mathbb{O}$ is not a ring	Octonion Algebra
the sedenions $\mathbb{S}$	a sixteen-dimensional real algebra with zero divisors, whose multiplication is not associative; neither a ring nor a division algebra	Division Algebras
the zero ring $\{0\}$	a ring, but excluded from every list by the corpus convention $1 \neq 0$	Rings, §2
a Lie algebra $\mathrm{G}$	not a ring: the bracket is bilinear and antisymmetric but not associative	Lie Algebras

Summary

This article has listed the rings of the corpus with their units, their zero divisors, their ideals and their chain conditions. The number systems $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, the split-complex numbers, the dual numbers, the quaternions, the split-biquaternions and the biquaternions open the list; the residue rings, the finite fields and the products follow; the polynomial, power series, localisation, quadratic-integer and Dedekind rings occupy the commutative middle; and the matrix, group, Weyl, free, tensor, exterior, Clifford and endomorphism rings close it. A final section gathers the rings the corpus meets outside its algebra articles: the Boolean rings, the cohomology and representation rings, the Grothendieck–Witt and Witt rings, the symmetric functions and the graded ring of modular forms. Beside the examples stand the non-examples — $\mathbb{Z}/6\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ among the finite rings, $\mathbb{Z}[\sqrt{-5}]$ and $\overline{\mathbb{Z}}$ among the commutative ones, $M_n(\mathbb{R})$ among the non-commutative ones — each with the failure named.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following, and they are the symbols of the articles that introduce them.

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
$\mathbb{D}$, $\mathbb{D}'$	Split-complex numbers and dual numbers
$\mathbb{H}$, $\mathbb{H}_{\mathbb{D}}$, $\mathbb{B}$, $\mathbb{O}$	Quaternions, split-biquaternions, biquaternions, octonions
$R[x]$, $R[x_1,\dots,x_n]$, $R[[x]]$	Polynomial and power series rings
$k(x)$, $k((t))$	Rational function field, Laurent series field
$\mathbb{Z}_{(p)}$	Localisation of $\mathbb{Z}$ at the prime $p$
$\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{2}]$, $\mathbb{Z}[\sqrt{-5}]$	Quadratic integer rings
$\mathcal{O}_K$, $\overline{\mathbb{Z}}$	Ring of integers of a number field, all algebraic integers
$M_n(R)$	Matrix ring
$k[G]$	Group algebra
$A_1(k)$, $D_1(k)$	The Weyl algebra $k\langle x,y\rangle/(yx-xy-1)$ and its division ring of fractions
$R\langle x_1,\dots,x_n\rangle$, $T(V)$	Free algebra and tensor algebra
$\Lambda(M)$, $\mathbb{C}\ell(V,q)$	Exterior algebra and Clifford algebra
$\operatorname{End}_A(M)$	Endomorphism ring of a module
$B$	A Boolean ring
$H^*(X;R)$, $R_k(G)$, $GW(F)$, $W(F)$	Cohomology ring, representation ring, Grothendieck–Witt ring, Witt ring
$\Lambda$, $M_*(\Gamma(1))$	Ring of symmetric functions, graded ring of modular forms