List of Non-Examples and Counter-Examples in Algebra

Introduction

This article lists the objects of the corpus that fail exactly one axiom, or exactly one rung of one of the algebra ladders, and it names the axiom each one fails. The five failures the corpus uses most often are the ones the scope of this list opens with: $\mathbb{Z}[\sqrt{-5}]$, which is a domain but not a unique factorisation domain; $\mathbb{Z}[x]$, which is a unique factorisation domain but not a principal ideal domain; the split-complex numbers and the dual numbers, which are not domains; $M_2(\mathbb{R})$, which is not a division ring; and the octonions, whose multiplication is not associative.

Every entry points to the article that records the object and the failure. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. The article is made of examples and non-examples side by side; here, indeed, the non-examples are the substance of the list, and each is paired with the example above it on the ladder, so that the reader sees the property that fails and the object that has it.

Not Domains: Zero Divisors

A domain is a commutative ring with $1 \neq 0$ and no zero divisors; the objects listed here fail that condition, each with an explicit pair of nonzero elements whose product is zero or a nonzero nilpotent element.

Object	The property above it, and the failure	Introduced in
Split-complex numbers $\mathbb{D}$	not a domain: $\pi_+ \pi_- = 0$ for $\pi_\pm = \tfrac12(1\pm t)$, both nonzero	Split-Complex Algebra
Dual numbers $\mathbb{D}'$	not a domain: $\varepsilon^2 = 0$ with $\varepsilon \neq 0$	Dual Numbers Algebra
$\mathbb{Z}/n\mathbb{Z}$ for composite $n$	not a domain: the residue classes of the proper factors of $n$ multiply to $0$	Examples of Rings and Fields
$\mathbb{Z}/4\mathbb{Z}$	not reduced: the class of $2$ is a nonzero nilpotent	Reduced Rings and the Nilradical
The exterior algebra $\Lambda(V)$, $\dim V \geq 1$	not a domain: $v \wedge v = 0$ for $v \in V$	The Exterior Algebra
Split biquaternions $\mathbb{H}_{\mathbb{D}}$	not a domain: they are $\mathbb{H}\oplus\mathbb{H}$, and have zero divisors	Split-Biquaternion Algebra
Biquaternions $\mathbb{B}$	not a domain: $\mathbb{B} \cong M_2(\mathbb{C})$ has zero divisors	Biquaternion Algebra
Sedenions $\mathbb{S}$	not a domain: the norm is not multiplicative and there are zero divisors	Octonion Algebra
The ring $\mathbb{Z}[\sqrt{-5}]$ as a counter-example	it is nevertheless a domain: it is the failure of unique factorisation, not of the domain axiom	Unique Factorisation Domains

Not Reduced: Nilpotents

A ring is reduced when its only nilpotent element is $0$; the objects here carry a nonzero nilpotent element, or a nonzero nilpotent ideal, and each row names the nilpotent that fails the condition.

Object	The property above it, and the failure	Introduced in
$\mathbb{Z}/4\mathbb{Z}$	not reduced: $2^2 = 0$	Reduced Rings and the Nilradical
$\mathbb{Z}/p^n\mathbb{Z}$, $n \geq 2$	not reduced: the class of $p$ is nilpotent	Reduced Rings and the Nilradical
$\mathbb{D}' = \mathbb{R}[\varepsilon]/(\varepsilon^2)$	not reduced: $\varepsilon^2 = 0$	Dual Numbers Algebra
$\mathbb{D}'_F = F[\varepsilon]/(\varepsilon^n)$	not reduced: $\varepsilon$ is nilpotent of index $n$	Noetherian and Artinian Rings
A degenerate Clifford algebra	not semisimple: the radical is a nonzero nilpotent ideal	Degenerate Clifford Algebras and the Radical
Non-example: $\mathbb{Z}/6\mathbb{Z}$ as non-reduced	it is not: $\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ is reduced, a product of fields	Reduced Rings and the Nilradical

Not Unique Factorisation Domains

A unique factorisation domain is a domain in which every nonzero nonunit is a product of irreducibles, uniquely up to order and associates, equivalently a domain in which every irreducible is prime. The standard failure is $\mathbb{Z}[\sqrt{-5}]$, where $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$ exhibits two factorisations.

Object	The property above it, and the failure	Introduced in
$\mathbb{Z}[\sqrt{-5}]$	a domain and a Dedekind domain, but not a unique factorisation domain: two factorisations of $6$	Unique Factorisation Domains
The class number as witness	the failure of unique factorisation is the nontriviality of the class group	Dedekind Domains and Ideal Class Groups
The ideal $(2, 1+\sqrt{-5})$	a nonprincipal ideal whose square is principal; the obstruction made explicit	Dedekind Domains and Ideal Class Groups
$\overline{\mathbb{Z}}$, the ring of algebraic integers	not a unique factorisation domain, and not Noetherian	Bézout Domains
Non-example: $\mathbb{Z}[\sqrt{-5}]$ as a non-domain	it is a domain: the failure is the factorisation, not the zero-divisor axiom	Integral Domains
Non-example: $\mathbb{Z}[\sqrt{-5}]$ as non-Noetherian	it is Noetherian: it is a Dedekind domain	Dedekind Domains and Ideal Class Groups

Not Principal Ideal Domains

A principal ideal domain is a domain in which every ideal is generated by one element; it is a unique factorisation domain, so a failure of unique factorisation is also a failure of principality, but the corpus's sharp examples keep the two failures apart.

Object	The property above it, and the failure	Introduced in
$\mathbb{Z}[x]$	a unique factorisation domain but not principal: the ideal $(2,x)$ is not generated by one element	Principal Ideal Domains
$k[x_1,\dots,x_n]$, $n \geq 2$	a unique factorisation domain but not principal: the ideal $(x_1,x_2)$	Principal Ideal Domains
$\overline{\mathbb{Z}}$	a Bézout domain but not Noetherian, hence not a principal ideal domain	Bézout Domains
$\mathbb{Z}[\sqrt{-5}]$	not principal, because not a unique factorisation domain	Unique Factorisation Domains
$\mathbb{Z}\left[\tfrac{1+\sqrt{-19}}{2}\right]$	a principal ideal domain that is not a Euclidean domain	Euclidean Domains
Non-example: $\mathbb{Z}$ as a non-principal domain	it is principal: every ideal is $(n)$	Principal Ideal Domains
Non-example: $k[x,y]$ as a Bézout domain	it is not: its finitely generated ideal $(x,y)$ is not principal	Bézout Domains

Not Euclidean Domains

A Euclidean domain is a domain with a degree function and division with remainder; every Euclidean domain is a principal ideal domain, and the failure here is the converse: a principal ideal domain that admits no Euclidean degree.

Object	The property above it, and the failure	Introduced in
$\mathbb{Z}\left[\tfrac{1+\sqrt{-19}}{2}\right]$	a principal ideal domain but not Euclidean; the standard counter-example	Euclidean Domains
Non-example: $\mathbb{Z}[i]$ as non-Euclidean	it is Euclidean, with the norm as degree function	Examples of Rings and Fields
Non-example: $\mathbb{Z}$ as a non-Euclidean domain	it is Euclidean: it is norm-Euclidean with the absolute value as norm	Euclidean Domains

Not Division Rings or Division Algebras

A division ring is a ring in which every nonzero element is a unit; a division algebra is the corresponding algebra over a field. The failures are of two kinds: the objects with zero divisors, and the objects that are division algebras but not rings because associativity fails.

Object	The property above it, and the failure	Introduced in
$M_2(\mathbb{R})$	a simple algebra but not a division ring: the matrix with one nonzero entry is a nonzero zero divisor	Examples of Algebras
Split biquaternions $\mathbb{H}_{\mathbb{D}}$	not a division algebra: $\cong \mathbb{H}\oplus\mathbb{H}$, with zero divisors and no nonzero nilpotents	Split-Biquaternion Algebra
Biquaternions $\mathbb{B}$	not a division ring: $\cong M_2(\mathbb{C})$	Biquaternion Algebra
$\mathbb{Z}$	a domain but not a division ring: only $\pm 1$ are units	The Integers
The free algebra $k\langle x_1,\dots,x_n\rangle$	a domain but not a division ring, and not commutative	Tensor Powers and the Free Algebra
The Weyl algebra $A_1$	a domain but not a division ring; it embeds in a division ring of fractions	Quotients of the Tensor Algebra
The octonions $\mathbb{O}$	a division algebra but not a ring: the multiplication is not associative	Octonion Algebra
The sedenions $\mathbb{S}$	not even a division algebra: zero divisors	Octonion Algebra
Non-example: $\mathbb{H}$ as a division ring that is not a field	it is a division ring: the failure is commutativity, not inverses	Quaternion Algebra
Non-example: a finite division ring that is not a field	does not exist, by Wedderburn's little theorem	Division Algebras

Not Fields

A field is a commutative division ring; the objects here are domains or division rings that fail one of the two clauses, either commutativity or the existence of inverses for every nonzero element.

Object	The property above it, and the failure	Introduced in
$\mathbb{H}$	a division ring but not a field: not commutative	Quaternion Algebra
$\mathbb{O}$	neither commutative nor associative, so neither a ring nor a field	Octonion Algebra
$\mathbb{Z}$	an integral domain but not a field: $2$ is not a unit	The Integers
$\mathbb{Z}_p$, the $p$-adic integers	a principal ideal domain but not a field: $p$ is not a unit	Absolute Values, Valuations and Completions
$k[x]$, $k[[x]]$	principal ideal domains but not fields: $x$ is not a unit	Polynomial Rings and Rational Functions
$\mathbb{Z}[\sqrt{-5}]$	a domain but not a field, and not a unique factorisation domain	Unique Factorisation Domains
The zero ring	not a field: $1 = 0$	Rings
Non-example: $\mathbb{D}$ and $\mathbb{D}'$ as fields	fail already the domain axiom: a field is a domain	Split-Complex Algebra
Non-example: $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{F}_p$ as non-fields	they are fields: the failures above are not generic	Fields

Not Associative and Not Commutative

Associativity and commutativity are the two axioms that survive the doubling chain and fail one at a time; the objects here each fail exactly one of them.

Object	The property above it, and the failure	Introduced in
The octonions $\mathbb{O}$	not associative: $(e_1e_2)e_4 = e_7$ but $e_1(e_2e_4) = -e_7$; alternative and power-associative	Octonion Algebra
The sedenions $\mathbb{S}$	neither associative nor alternative, and not a division algebra	Octonion Algebra
A Lie algebra $\mathrm{G}$	not associative and not commutative: $[x,y] = -[y,x]$	Lie Algebras
A Jordan algebra	not associative: the Jordan identity replaces associativity	Jordan Algebras
The free algebra $k\langle x_1,\dots,x_n\rangle$	associative but not commutative for $n \geq 2$	Tensor Powers and the Free Algebra
The exterior algebra $\Lambda(V)$	associative but graded-commutative, not commutative for $\dim V \geq 2$	The Exterior Algebra
$M_2(\mathbb{R})$	associative but not commutative	Examples of Algebras
$\mathbb{H}$	associative but not commutative	Quaternion Algebra
Non-example: $\mathbb{C}$ as non-commutative	it is commutative: commutativity is lost only at $\mathbb{H}$	The Complex Numbers

Not Semisimple

A semisimple algebra is a product of matrix algebras over division rings; by Wedderburn–Artin it is exactly an algebra whose radical vanishes. The failures here have a nonzero radical, and the group algebra in the modular case is the classical one.

Object	The property above it, and the failure	Introduced in
$k[G]$ with $\operatorname{char} k$ dividing $\|G\|$	not semisimple: Maschke's theorem fails, and the radical is nonzero	Group Algebras
The modular group algebra	the failure of Maschke's theorem and the decomposition of the regular representation	Modular Representation Theory
$k[x]/(x^2)$	not semisimple: the radical is $(x)$	Simple and Semisimple Modules
A degenerate Clifford algebra	not semisimple: the radical is nonzero	Degenerate Clifford Algebras and the Radical
$\mathbb{Z}/4\mathbb{Z}$	not semisimple as a ring over itself: the radical is $(2)$	Noetherian and Artinian Rings
The dual numbers $\mathbb{D}'$	not semisimple: local with a nilpotent maximal ideal	Dual Numbers Algebra
Non-example: a semisimple algebra over a field of characteristic $0$	it is semisimple: Maschke's theorem holds when the characteristic does not divide the group order	Group Algebras

Summary

The list gathers the objects of the corpus that fail exactly one axiom. The failures of the multiplicative structure are the zero divisors of the split-complex numbers, the dual numbers, $\mathbb{Z}/n\mathbb{Z}$ for composite $n$, the exterior algebra and the split-biquaternions; the nilpotents of $\mathbb{Z}/4\mathbb{Z}$, the dual numbers and a degenerate Clifford algebra; the matrix algebra $M_2(\mathbb{R})$, the biquaternions and the sedenions as non-division rings; and $\mathbb{Z}$, $\mathbb{Z}_p$, $k[x]$ and $\mathbb{Z}[\sqrt{-5}]$ as non-fields. The failures of the divisibility ladders are $\mathbb{Z}[\sqrt{-5}]$, which is not a unique factorisation domain, $\mathbb{Z}[x]$ and $k[x_1,\dots,x_n]$, which are not principal ideal domains, and $\mathbb{Z}\left[\tfrac{1+\sqrt{-19}}{2}\right]$, which is a principal ideal domain that is not Euclidean. The failures of the structural axioms are the octonions and the sedenions, which are not associative, the free algebra and the exterior algebra, which are not commutative, and the modular group algebra, which is not semisimple. Each row names the example above the failure, so that the strictness of the ladder is visible in the list itself.

Summary of Notation

The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.

Symbol	Meaning
$\mathbb{Z}[\sqrt{-5}]$, $\overline{\mathbb{Z}}$	the standard non-unique-factorisation domain, the algebraic integers
$\mathbb{Z}[x]$, $k[x_1,\dots,x_n]$	polynomial rings, unique factorisation domains that are not principal
$\mathbb{D}$, $\mathbb{D}'$	split-complex numbers, dual numbers
$M_2(\mathbb{R})$, $\mathbb{B}$	matrix algebra, biquaternions
$\mathbb{H}$, $\mathbb{O}$, $\mathbb{S}$	quaternions, octonions, sedenions
$\Lambda(V)$, $k\langle x_1,\dots,x_n\rangle$	exterior algebra, free algebra
$k[G]$	group algebra
$\mathbb{Z}/n\mathbb{Z}$	residue rings
$\mathbb{Q}$, $\mathbb{C}$, $\mathbb{F}_p$	the rationals, the complex numbers and the prime fields