List of Non-Commutative Rings

Introduction

This article lists the non-commutative rings of the corpus — the matrix rings, the quaternions and their relatives, the group rings, the free algebra, the tensor algebra and the Weyl algebra — and it records what is lost by comparison with the commutative case. Every entry points to the article that introduces the object.

The list gathers the rings in which $ab \neq ba$ for some pair of elements. It includes the division ring $\mathbb{H}$, which is the non-commutative ring with the fewest zero divisors; the matrix rings, which are the non-commutative source of idempotents and one-sided ideals; the group rings, which are non-commutative exactly when the group is; the free algebra and the tensor algebra, which are the freely generated cases; and the Weyl algebra, which is the quantised polynomial ring.

The article introduces nothing and proves nothing. It records examples and non-examples side by side, and it names the commutative features that each non-commutative ring replaces.

The Matrix Rings

Ring	Centre	Zero divisors	The structure that is lost	Introduced in
$M_n(R)$, $n \geq 2$	the scalars	$E_{11}E_{22} = 0$	commutativity; the ring is simple for $R$ a field	Matrix Algebras
$M_2(\mathbb{C})$	$\mathbb{C}$	yes	the biquaternion algebra is this ring	Biquaternion Algebra
the upper triangular matrices	the diagonal	strictly upper triangular matrices square to zero	not simple: it has a nontrivial nilpotent ideal	Matrix Algebras
$M_n(R) \times M_m(S)$	$Z(R) \times Z(S)$	$(1,0)(0,1) = 0$	not simple	Examples of Rings and Fields

The matrix ring is the standard non-commutative ring: it is simple when the coefficient ring is a field, so it has no nontrivial two-sided ideals, and it is therefore not a domain and not a division ring, since the matrix units are zero divisors. The Wedderburn–Artin theorem writes every finite-dimensional semisimple algebra as a product of matrix algebras over division algebras, and this is recorded in Division Algebras.

The Quaternion Algebras and Their Relatives

Ring	Centre	Zero divisors	What it is	Introduced in
$\mathbb{H}$	$\mathbb{R}$	none	a division ring, and a domain	Quaternion Algebra
$\mathbb{B}$, the biquaternions	$\mathbb{C}$	yes	isomorphic to $M_2(\mathbb{C})$ as a $\mathbb{C}$-algebra	Biquaternion Algebra
$\mathbb{H}_{\mathbb{D}}$, the split-biquaternions	$\mathbb{D}$	yes	isomorphic to $\mathbb{H} \oplus \mathbb{H}$ via the idempotents $e_{\pm}$	Split-Biquaternion Algebra
$\mathbb{O}$, the octonions	$\mathbb{R}$	none	not a ring: the multiplication is not associative	Octonion Algebra

The three eight-dimensional relatives of $\mathbb{H}$ differ exactly in the sign of the norm, and the sign decides whether a zero divisor exists. The biquaternions have the algebraically closed centre $\mathbb{C}$ and split as $M_2(\mathbb{C})$; the split-biquaternions have the split centre $\mathbb{D} = \mathbb{R}[j]/(j^2-1)$ and split as $\mathbb{H} \oplus \mathbb{H}$ through the idempotents $e_{\pm} = \tfrac{1}{2}(1 \pm j)$, which satisfy $\pi_+ \pi_- = 0$. The octonions are excluded from the list of rings by non-associativity and not by a zero divisor.

The Group Rings

Group ring	The group	Commutative?	Zero divisors	Introduced in
$k[G]$ for $G$ abelian	abelian	yes, a commutative ring	none if $G$ is torsion-free and $k$ a domain	Group Algebras
$k[G]$ for $G$ non-abelian	non-abelian	no	none if $G$ is torsion-free and $k$ a domain	Group Algebras
$k[F_n]$, the free group	$F_n$	no for $n \geq 2$	none: $F_n$ is torsion-free	Ore Domains and Division Rings of Fractions
$\mathbb{Q}[C_3]$	$C_3$	yes	$\mathbb{Q}[C_3] \cong \mathbb{Q} \times \mathbb{Q}(\zeta_3)$, so it has idempotents	Examples of Rings and Fields
$\mathbb{F}_2[C_2]$	$C_2$	yes	$\mathbb{F}_2[C_2] \cong \mathbb{F}_2[x]/(x+1)^2$, with a nilpotent	Examples of Rings and Fields
$k[C_p]$, with $\zeta_p \in k$ and $\operatorname{char} k \nmid p$	$C_p$	yes	$k[C_p] \cong k^p$, a product of fields	Group Algebras

A group ring is commutative exactly when the group is abelian, so the non-commutative group rings are those of the non-abelian groups. The torsion-free hypothesis governs the zero divisors and not the commutativity: $k[F_n]$ is a non-commutative domain, while $k[C_2]$ is a commutative ring with zero divisors. The group ring of a torsion-free group is a domain, and it is listed among the non-commutative domains in List of Domains.

The Free Algebra, the Tensor Algebra and the Weyl Algebra

Ring	Presentation	The universal property	Introduced in
$R\langle x_1, \dots, x_n\rangle$, the free algebra	the non-commuting polynomial ring	the free $R$-algebra on $n$ generators	Tensor Powers and the Free Algebra
$T(V)$, the tensor algebra	$\bigoplus_{n \geq 0} V^{\otimes n}$	the free associative algebra on the vector space $V$	Tensor Powers and the Free Algebra
$\Lambda(V)$, the exterior algebra	$T(V)/(v \otimes v)$	the free graded-commutative algebra; $v^2 = 0$	The Exterior Algebra
$A_1(k) = k\langle x, y\rangle/(yx - xy - 1)$, the Weyl algebra	the quantised polynomial ring in two variables	the algebra of differential operators on $k[x]$	Quotients of the Tensor Algebra
$\mathbb{C}\ell(V, q)$, a Clifford algebra	$T(V)/(v \otimes v - q(v))$	the quantised exterior algebra	Clifford Algebras

The free algebra and the tensor algebra are the non-commutative analogues of the polynomial ring, and they lose the commutative universal property: a homomorphism from $R\langle x_1, \dots, x_n\rangle$ may send the $x_i$ to arbitrary elements, but the images need not commute, so the commutative polynomial ring is a quotient of the free algebra by the commutator ideal. The Weyl algebra and the Clifford algebras are the quotients of the tensor algebra in which the commutator and the square are fixed by a relation, and they are the standard quantisations.

The Two-Sided Ideals

Ring	Simple?	The two-sided ideals	Introduced in
$M_n(k)$, $n \geq 2$	yes	$(0)$ and $M_n(k)$ alone	Matrix Algebras
$\mathbb{H}$	yes	$(0)$ and $\mathbb{H}$ alone	Quaternion Algebra
$\mathbb{B} \cong M_2(\mathbb{C})$	yes	$(0)$ and $\mathbb{B}$ alone	Biquaternion Algebra
$\mathbb{H}_{\mathbb{D}} \cong \mathbb{H} \oplus \mathbb{H}$	no	the two ideals $\mathbb{H}\pi_+$ and $\mathbb{H}\pi_-$	Split-Biquaternion Algebra
$A_1(k)$	yes	$(0)$ and $A_1(k)$ alone	Quotients of the Tensor Algebra
the free algebra $R\langle x_1, \dots, x_n\rangle$	no	the commutator ideal and the ideals generated by the relations	Tensor Powers and the Free Algebra
the upper triangular matrices	no	the nilpotent ideal of strictly upper triangular matrices	Matrix Algebras

The simple rings replace the maximal ideals of the commutative theory: a commutative ring is a field exactly when its only ideals are $(0)$ and $(1)$, and the non-commutative analogue is the simple ring, of which $M_n(k)$ and $A_1(k)$ are the standard examples. The split-biquaternions are the example that shows the difference between simple and semisimple, since they are semisimple without being simple, and their two ideals come from the idempotents $e_{\pm}$.

What Is Lost by Comparison with the Commutative Case

Commutative feature	What replaces it	The witness	Introduced in
$ab = ba$	the commutator $[a,b] = ab - ba$	$M_2(\mathbb{R})$ and every matrix ring	Matrix Algebras
left ideals are right ideals	the two-sided, left and right ideals differ	the free algebra	Non-Commutative Domains
the field of fractions always exists	a division ring of fractions exists only under the Ore condition	$A_1(k)$ is Ore, the free algebra is not	Ore Domains and Division Rings of Fractions
every finite domain is a field	Wedderburn's little theorem: every finite division ring is a field	the finite fields of List of Finite Fields	Division Rings
the maximal ideals give quotients that are fields	simple rings replace the maximal ideals	$M_n(\mathbb{R})$ is simple, and not a field	Matrix Algebras

The losses accumulate from the first row to the last: the commutative theory of divisibility rests on the maximal ideals and the primes, and the non-commutative theory replaces them with the simple rings and the one-sided ideals. Every row of the table is a feature that a reader arriving from the commutative sections of this Part will look for in vain, and the last column names where the non-commutative substitute is developed.

Warnings

Object	Why it is not a non-commutative ring of this list	Introduced in
$\mathbb{O}$, the octonions	not associative, so not a ring	Octonion Algebra
$\mathbb{S}$, the sedenions	not associative, and with zero divisors	Division Algebras
$\mathbb{D}$, the split-complex numbers	commutative, with zero divisors	Split-Complex Algebra
$\mathbb{D}'$, the dual numbers	commutative, with a nilpotent	Dual-Numbers Algebra
$\mathbb{Z}/6\mathbb{Z}$	commutative, with zero divisors	Modular Arithmetic and the Ring of Residues
every commutative ring	not on this list: commutativity is the criterion for exclusion	Commutative Rings

The objects in the first two rows are the ones a reader might expect here on the strength of the quaternions: they are the next two algebras of the Cayley–Dickson chain, and they are excluded by the failure of associativity at the octonions.

Summary

This article has listed the non-commutative rings of the corpus: the matrix rings $M_n(R)$, which are simple and have zero divisors; the quaternions, which are a division ring; the biquaternions and split-biquaternions, which are the eight-dimensional relatives with zero divisors and the identifications $M_2(\mathbb{C})$ and $\mathbb{H} \oplus \mathbb{H}$; the group rings, non-commutative exactly for the non-abelian groups and a domain for the torsion-free ones; the free algebra and the tensor algebra, which are the freely generated cases; and the Weyl algebra and the Clifford algebras, which are the quantised quotients. Beside them the article has recorded what the commutative case loses: commutativity replaced by the commutator, the coincidence of left and right ideals lost, the field of fractions made conditional on the Ore condition, the maximal ideals replaced by the simple rings, and unique factorisation without a general theory.

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
$M_n(R)$	The $n \times n$ matrix ring
$E_{11}, E_{22}$	Matrix units
$\mathbb{H}$, $\mathbb{B}$, $\mathbb{H}_{\mathbb{D}}$, $\mathbb{O}$	Quaternions, biquaternions, split-biquaternions, octonions
$\mathbb{D}$	The split-complex numbers, the centre of $\mathbb{H}_{\mathbb{D}}$
$e_{\pm} = \tfrac{1}{2}(1 \pm j)$	The idempotents of $\mathbb{H}_{\mathbb{D}}$
$k[G]$, $k[F_n]$, $k[C_2]$	Group rings; $F_n$ free, $C_2$ of order two
$R\langle x_1,\dots,x_n\rangle$	The free algebra
$T(V)$, $\Lambda(V)$, $\mathbb{C}\ell(V,q)$	Tensor, exterior and Clifford algebras
$A_1(k)$	The Weyl algebra $k\langle x,y\rangle/(yx - xy - 1)$
$[a,b] = ab - ba$	The commutator