List of Algebraic Structures

Introduction

This article lists the algebraic structures the corpus builds, one rung at a time, and the axiom each rung adds. The ladder begins with a set carrying a single binary operation and climbs by adding associativity, an identity, inverses and commutativity; a second operation then makes a ring and, once every nonzero element becomes invertible, a field; scalars acting on an abelian group make a module, a vector space and finally an algebra. Some rungs are non-associative rather than associative — the Lie algebras and the Jordan algebras — and they are gathered apart from the associative ladder.

Every entry points to the article that introduces the structure and states the property that fixes the rung. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being an object that differs from its rung by exactly one axiom, with the failure named and the article that records it.

The One-Operation Rungs

A monoid is the lowest rung the corpus writes down: one associative operation and an identity, with no assumption of inverses or commutativity. A group adds inverses, and an abelian group adds commutativity on top of the group axioms. The scope of the list names a closure-only rung and a semigroup rung below the monoid; neither is introduced anywhere in Parts I to III, and no entry is written for them.

Structure	The property it has	Introduced in
Monoid	one associative binary operation with an identity	Universal Properties and Categories
Group	a monoid in which every element is invertible	Groups, §1
Abelian group	a group whose operation is commutative	Groups, §1
Non-example: $(\mathbb{Z} \setminus \{0\}, \cdot)$	a monoid that fails inverses; only $\pm 1$ are invertible	Groups, §1
Non-example: the symmetric group $S_n$, $n \geq 3$	a group that fails commutativity	Groups, §1
Non-example: $\mathbb{N}$ under addition	a commutative monoid that fails additive inverses	The Natural Numbers (Part V)

The Two-Operation Rungs

A ring is an abelian group carrying a second operation, a multiplication that is associative and distributive over the addition; the corpus fixes $1 \neq 0$. The commutative ring adds $ab = ba$, the integral domain adds the absence of zero divisors, and the field requires every nonzero element to be invertible. The division ring keeps associativity and inverses but drops commutativity; it is the rung at which the commutative and the non-commutative ladders part.

Structure	The property it has	Introduced in
Ring	an abelian group with an associative multiplication distributive over the addition	Rings, §1
Commutative ring	a ring whose multiplication is commutative	Commutative Rings; stated in Rings, §5
Reduced ring	a commutative ring with no nonzero nilpotent	Reduced Rings and the Nilradical
Integral domain	a commutative ring with $1 \neq 0$ and no zero divisors	Integral Domains; stated in Rings, §12
GCD domain	a domain in which any two elements have a greatest common divisor	GCD Domains
Bézout domain	a domain in which every finitely generated ideal is principal	Bézout Domains
Unique factorisation domain	a domain in which every nonzero nonunit factors uniquely into irreducibles	Unique Factorisation Domains
Principal ideal domain	a domain in which every ideal is principal	Principal Ideal Domains
Euclidean domain	a domain carrying a Euclidean degree function and division with remainder	Euclidean Domains
Division ring	a ring with $1 \neq 0$ in which every nonzero element is invertible	Division Rings
Ore domain	a domain in which every two nonzero elements have a common left and a common right multiple	Ore Domains and Division Rings of Fractions
Field	a commutative ring with $1 \neq 0$ in which every nonzero element is invertible	Fields, §1
Non-example: $\mathbb{Z}$	an integral domain that fails the field axiom: $2$ has no inverse	The Integers (Part V); Rings
Non-example: the split-complex numbers $\mathbb{D}$	a commutative ring that fails the domain axiom: $(1+j)(1-j) = 0$	Examples of Rings and Fields
Non-example: the dual numbers $\mathbb{D}'$	a commutative ring that fails the domain axiom: $\varepsilon^2 = 0$	Examples of Rings and Fields
Non-example: the matrix ring $M_2(\mathbb{R})$	a ring that fails commutativity and fails inverses for nonzero elements	Examples of Algebras
Non-example: the quaternions $\mathbb{H}$	a division ring that fails commutativity, hence is not a field	Division Algebras

The Scalar-Action Rungs

A module is an abelian group on which a ring acts by scalars, and a vector space is the case of a field of scalars, where every module is free and every space has a basis. An algebra adds a bilinear product of the space with itself. The corpus's broad sense of algebra assumes neither associativity, nor commutativity, nor a unit; the qualifiers are earned separately.

Structure	The property it has	Introduced in
Module over a ring $R$	an abelian group with a scalar action of $R$	Modules
Vector space over a field	a module over a field, hence free with a basis	Vector Spaces
Algebra over a commutative ring	a module with a bilinear product	Algebras
Associative algebra	an algebra with $(xy)z = x(yz)$	Algebras
Unital algebra	an algebra with an identity for its product	Algebras
Commutative algebra	an associative algebra with $xy = yx$	Commutative Algebras
Ideals and quotients of an algebra	the two-sided ideals of an algebra and the quotient algebras they define	Ideals and Quotients of Algebras
Centre, units and zero divisors	the centre $Z(A)$, the group of units $A^\times$ and the zero divisors of an algebra	Centre, Units, Zero Divisors and Division Algebras

The Non-Associative Rungs

Two rungs replace associativity by an identity of their own: the Lie algebra, whose product is antisymmetric and satisfies the Jacobi identity, and the Jordan algebra, whose product is commutative and satisfies the Jordan identity. The non-associative algebras in general, and the identities that lie between associativity and its failure, are the subject of Non-Associative Algebras and the Property Ladder. The octonions are the standard non-example: a division algebra whose multiplication is not associative.

Structure	The property it has	Introduced in
Lie algebra	an antisymmetric product satisfying the Jacobi identity	Lie Algebras
Jordan algebra	a commutative product satisfying the Jordan identity	Jordan Algebras
Non-associative algebra	an algebra carrying no associativity axiom	Non-Associative Algebras and the Property Ladder
Non-example: the octonions $\mathbb{O}$	a division algebra that fails associativity, hence is not a ring	Division Algebras; Octonion Algebra (Part V)
Non-example: the cross product on $\mathbb{R}^3$	an algebra that fails associativity and has no identity	Algebras

Structures Defined by an Order

The corpus also meets two rungs whose operation is not written as a multiplication: the lattice, with its two order-theoretic operations of join and meet, and the Boolean algebra of the algebra of subsets and of propositional logic. The lattice is introduced in the foundational layer; the Boolean algebra is developed as a subject in the synthetic study of the Booleans.

Structure	The property it has	Introduced in
Lattice	a partial order in which every two elements have a join and a meet	Order Theory and Lattices
Complete lattice	a lattice in which every subset has a supremum and an infimum	Order Theory and Lattices
Boolean algebra	a complemented distributive lattice, the algebra of subsets	Boolean Algebras and Lattices (Part V)
Non-example: the lattice $N_5$	a modular lattice that fails distributivity	Order Theory and Lattices

The Axiom Added at Each Rung

The ladder is a list of one added axiom per rung, and the table reads it in that direction: each row names the axiom the rung adds to the one above it, with an object at the rung.

Rung	The axiom it adds to the rung above	A structure at the rung	Introduced in
Monoid	associativity and an identity, from a bare operation	$(\mathbb{N}, +)$	Universal Properties and Categories
Group	inverses	$(\mathbb{Z}, +)$	Groups, §1
Abelian group	commutativity	$(\mathbb{Z}, +)$	Groups, §1
Ring	a second operation, distributing over the first	$\mathbb{Z}$	Rings, §1
Commutative ring	commutativity of the second operation	$\mathbb{Z}$	Rings, §5
Integral domain	no zero divisors	$\mathbb{Z}$	Rings, §12
Field	inverses for the nonzero elements	$\mathbb{Q}$	Fields, §1
Module	an action of a ring on an abelian group	$\mathbb{Z}^n$	Modules
Vector space	the scalars form a field	$\mathbb{R}^n$	Vector Spaces
Algebra	the abelian group carries a bilinear product	$M_2(\mathbb{R})$	Algebras

The non-associative rungs branch off at the algebra: the Lie algebras and the Jordan algebras keep the module structure and change the identity the product satisfies, which is why they are listed apart from the associative ladder.

Summary

The list gathers the rungs of the algebraic ladder the corpus builds. The one-operation rungs are the monoid, the group and the abelian group; the two-operation rungs are the ring, the commutative ring, the reduced ring, the integral domain, the GCD, Bézout, unique factorisation, principal ideal and Euclidean domains, the division ring, the Ore domain and the field; the scalar-action rungs are the module, the vector space and the algebra with its associative, unital and commutative refinements; and the non-associative rungs are the Lie algebra, the Jordan algebra and the general non-associative algebra, with the lattice and the Boolean algebra met through an order rather than a product. Each row points to the article that introduces the structure, and the non-examples — $(\mathbb{Z} \setminus \{0\}, \cdot)$, $S_n$, $\mathbb{N}$, $\mathbb{Z}$, the split-complex numbers and the dual numbers, $M_2(\mathbb{R})$, $\mathbb{H}$, $\mathbb{O}$, the cross product on $\mathbb{R}^3$ and the lattice $N_5$ — name the single axiom each one fails.

Summary of Notation

The article denotes its objects by name; the symbols that appear in the tables are the standard number systems and the two two-dimensional algebras.

Symbol	Meaning
$\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$	the naturals, the integers, the rationals, the reals
$\mathbb{D}$	the split-complex numbers, unit $j$, $j^2 = +1$
$\mathbb{D}'$	the dual numbers, unit $\varepsilon$, $\varepsilon^2 = 0$
$\mathbb{H}$	the real quaternions
$\mathbb{O}$	the octonions
$S_n$	the symmetric group on $n$ letters
$M_2(\mathbb{R})$	the ring of $2 \times 2$ real matrices
$N_5$	the pentagon lattice, the non-distributive modular lattice
$Z(A)$, $A^\times$	the centre and the group of units of an algebra $A$
$1 \neq 0$	the nontriviality convention fixed for every ring and algebra