Examples of Algebras

Introduction

The preceding articles developed the general theory of algebras over a commutative ring: ideals and quotients, the centre, units and zero divisors, automorphisms and derivations, the tensor constructions, the topological layer and the operator-algebraic layer. This article collects the standard examples and records, for each, the data the theory associates with it — the dimension, the centre, the group of units, the zero divisors, the ideals, and the position of the algebra in the classification into division, simple and semisimple algebras. It is a reference article: the individual algebras are treated in their own articles, and the purpose here is to make the data comparable at a glance.

The ground ring is a field $k$ unless a different ring is stated, and the finite-dimensional algebras are over their ground field. The number systems are those of the shared conventions: $\mathbb{R}$, $\mathbb{C}$, the split complex numbers $\mathbb{D}$ with unit $j$, $j^2 = +1$, the dual numbers $\mathbb{D}'$ with unit $\varepsilon$, $\varepsilon^2 = 0$, the quaternions $\mathbb{H}$, the split biquaternions $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_\mathbb{R}\mathbb{H}$, and the biquaternions $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$.

The Data Table

The following table is the summary; each entry is justified in the sections below.

Algebra	$\dim_k$	Centre	Units	Zero divisors	Ideals	Type
$\mathbb{R}$	$1$	$\mathbb{R}$	$\mathbb{R}\setminus\{0\}$	none	$0,\ \mathbb{R}$	field, division
$\mathbb{C}$	$2$	$\mathbb{C}$	$\mathbb{C}\setminus\{0\}$	none	$0,\ \mathbb{C}$	field, division
$\mathbb{D}$	$2$	$\mathbb{D}$	$N(u) \neq 0$	yes	$0,\ \mathbb{D}e_\pm,\ \mathbb{D}$	semisimple, not simple
$\mathbb{D}'$	$2$	$\mathbb{D}'$	$x \neq 0$	yes	$(\varepsilon^m)$	local, not semisimple
$\mathbb{H}$	$4$	$\mathbb{R}$	$\mathbb{H}\setminus\{0\}$	none	$0,\ \mathbb{H}$	division, simple
$\mathbb{H}_{\mathbb{D}}$	$8$	$\mathbb{D}$	$N(u) \in \mathbb{D}^\times$	yes	two maximal ideals	semisimple, not simple
$\mathbb{B}$	$8$ (over $\mathbb{R}$)	$\mathbb{C}$	$N(\tilde{Q}) \neq 0$	yes	$0,\ \mathbb{B}$	simple, not division
$M_n(k)$, $n \geq 2$	$n^2$	$k I_n$	$\mathrm{GL}_n(k)$	yes	$0,\ M_n(k)$	simple, not division
$k[x]$	$\infty$	$k[x]$	$k^\times$	none	principal	domain, not simple
$k[G]$, $G$ finite	$\lvert G\rvert$	class sums	contains $G$	yes if $\lvert G\rvert\geq2$	from representations	semisimple if $\operatorname{char} k \nmid \lvert G\rvert$

The One-Dimensional and Two-Dimensional Algebras

The Real Numbers

$\mathbb{R}$ is the unique one-dimensional real algebra with $1 \neq 0$: the product is forced by bilinearity and $1\cdot 1 = 1$, and the basis element $1$ has square $+1$. It is a field, hence a division algebra; its only ideals are $0$ and $\mathbb{R}$; and its group of units is $\mathbb{R}^\times = \mathbb{R}\setminus\{0\}$.

The Complex Numbers

$\mathbb{C} = \mathbb{R}[x]/(x^2+1)$ is two-dimensional over $\mathbb{R}$, commutative, and a field; the generator $i$ satisfies $i^2 = -1$, and the only solution of $z z^* = 0$ is $z = 0$, so every nonzero element is a unit. Its only ideals are $0$ and $\mathbb{C}$. Over a general field $k$, the algebra $k[x]/(f)$ is a field exactly when $f$ is irreducible, which for $f = x^2+1$ happens over $k = \mathbb{R}$ and fails over $k = \mathbb{C}$; this is the simplest instance of the dependence of the structure on the ground field.

The Split Complex Numbers

$\mathbb{D} = \mathbb{R}[x]/(x^2-1)$ is two-dimensional and commutative, with unit $j$, $j^2 = +1$. The element $u = x + jy$ has

$$ N(u) = u u^* = x^2 - y^2, \qquad u^{-1} = \frac{x - jy}{x^2 - y^2} \ \ (x^2 \neq y^2), $$

so the units are the elements off the two lines $x = \pm y$, and the nonzero elements on those lines are zero divisors. The idempotents

$$ e_+ = \tfrac{1}{2}(1+j), \qquad e_- = \tfrac{1}{2}(1-j) $$

satisfy $e_+^2 = e_+$, $e_-^2 = e_-$, $e_+e_- = 0$, $e_+ + e_- = 1$, and exhibit the isomorphism $\mathbb{D} \cong \mathbb{R}\times\mathbb{R}$, $x + jy \mapsto (x+y, x-y)$. Through this isomorphism the ideals are $0$, $\mathbb{R}e_+$, $\mathbb{R}e_-$ and $\mathbb{D}$: the algebra is semisimple (it is a product of two fields) but not simple, and it is not a domain. The relation $e_+e_- = 0$ with both factors nonzero is the archetype of a zero-divisor pair.

The Dual Numbers

$\mathbb{D}' = \mathbb{R}[x]/(x^2)$ is two-dimensional and commutative, with unit $\varepsilon$, $\varepsilon^2 = 0$. The element $x + \varepsilon y$ is a unit exactly when $x \neq 0$, with

$$ (x + \varepsilon y)^{-1} = \frac{1}{x} - \frac{y}{x^2}\,\varepsilon, $$

and every element of the maximal ideal $(\varepsilon)$ is nilpotent, hence a zero divisor. The ideals of $\mathbb{D}'$ are the powers $(\varepsilon^m)$, $m = 0, 1, 2$, so the algebra is a local ring with unique maximal ideal $(\varepsilon)$; it is not semisimple, because its Jacobson radical $(\varepsilon)$ is nonzero and nilpotent. Unlike $\mathbb{D}$, the algebra $\mathbb{D}'$ cannot be decomposed as a product of fields, and the obstruction is exactly the nilpotent.

The Four-Dimensional and Higher Real Algebras

The Quaternions

$\mathbb{H}$ is four-dimensional over $\mathbb{R}$, with basis $e_0 = 1$, $e_1, e_2, e_3$, $e_k^2 = -e_0$ and $e_1e_2 = e_3$ cyclically. Every nonzero quaternion is a unit and $\mathbb{H}$ is a division algebra. Its centre is $\mathbb{R}\cdot 1$, it has no nonzero zero divisors, and its only ideals are $0$ and $\mathbb{H}$: it is a division ring, hence simple. It is non-commutative, so it is the first example in the list that is neither a field nor commutative.

The Split Biquaternions

$\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_\mathbb{R}\mathbb{H}$ is eight-dimensional over $\mathbb{R}$ and four-dimensional over the split complex numbers. Since $\mathbb{D} \cong \mathbb{R}\times\mathbb{R}$ as a real algebra, distributivity of the tensor product gives

$$ \mathbb{H}_{\mathbb{D}} \cong (\mathbb{R}\times\mathbb{R})\otimes_\mathbb{R}\mathbb{H} \cong \mathbb{H}\times\mathbb{H}. $$

So $\mathbb{H}_{\mathbb{D}}$ is a product of two copies of $\mathbb{H}$: it is semisimple, it is not simple, its centre is $\mathbb{D}$, and it has zero divisors, beginning with $1+j$ and $1-j$, which are the images of $(2,0)$ and $(0,2)$ and multiply to zero. This algebra is not the four-dimensional coquaternion algebra $M_2(\mathbb{R})$; the two are distinct, and the corpus's $\mathbb{H}_{\mathbb{D}}$ is the tensor product above.

The Biquaternion Algebra

$\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ is four-dimensional over $\mathbb{C}$ and eight-dimensional over $\mathbb{R}$. Its centre is the copy of $\mathbb{C}$ spanned by $e_0$.

Unlike the quaternion case, the element $i e_0 + e_1$ is a zero divisor, $(i e_0 + e_1)(i e_0 - e_1) = -1 + 1 = 0$. Thus $\mathbb{B}$ has zero divisors and is not a division algebra. As a $\mathbb{C}$-algebra it is isomorphic to $M_2(\mathbb{C})$, so it is simple, with only the trivial two-sided ideals, and its centre is $\mathbb{C}$. It is the standard example of a simple algebra that is not a division algebra, and the two phenomena — simplicity and zero divisors — coexist because the ground field is $\mathbb{C}$ rather than $\mathbb{R}$.

The General Framework

Two classical theorems organise the whole table, and the examples are best read as instances of them.

Theorem (Wedderburn–Artin, standard). A finite-dimensional semisimple algebra over a field $k$ is isomorphic to a finite product

$$ A \cong M_{n_1}(D_1) \times \dots \times M_{n_r}(D_r), $$

where the $D_i$ are finite-dimensional division algebras over $k$. The algebra is simple if and only if $r = 1$.

Theorem (Jacobson radical, standard). The radical $J(A)$ of a finite-dimensional $k$-algebra is a nilpotent two-sided ideal containing every nilpotent one-sided ideal, and $A$ is semisimple if and only if $J(A) = 0$. For a finite-dimensional algebra this is equivalent to the absence of a nonzero nilpotent two-sided ideal, and for a commutative algebra it is equivalent to the algebra being a product of fields.

These two theorems explain the entries of the table. The split complex numbers decompose as $\mathbb{D} \cong \mathbb{R}\times\mathbb{R}$, a product of two fields, hence $r = 2$, $n_1 = n_2 = 1$, $D_1 = D_2 = \mathbb{R}$; the split biquaternions decompose as $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\times\mathbb{H}$, hence $r = 2$ and $D_1 = D_2 = \mathbb{H}$; the biquaternions are $M_2(\mathbb{C})$, hence $r = 1$; and the dual numbers have $J(\mathbb{D}') = (\varepsilon)\neq0$, so they are not semisimple. The classification is not a mere list: each algebra is placed by its decomposition into matrix algebras over division algebras.

The Centre Computed

Three of the entries in the table are worth computing rather than quoting.

The centre of $\mathbb{H}$. Let $q = q_0 + q_1e_1 + q_2e_2 + q_3e_3$ commute with $e_1$. Then

$$ qe_1 = -q_1 + q_0e_1 - q_2e_3 + q_3e_2, \qquad e_1q = -q_1 + q_0e_1 + q_2e_3 - q_3e_2, $$

so $q_2 = q_3 = 0$. Commuting with $e_2$ in the same way gives $q_1 = q_3 = 0$. Hence $q = q_0 \in \mathbb{R}$, and $Z(\mathbb{H}) = \mathbb{R}$.

The centre of $M_n(k)$. If $A$ commutes with every matrix unit $E_{ij}$, then for every $i,j$,

$$ E_{ij}A = AE_{ij}, $$

which on comparing the two sides forces $A_{ii} = A_{jj}$ for all $i,j$ and $A_{rs} = 0$ for $r \neq s$. Hence $A = \lambda I_n$, and $Z(M_n(k)) = kI_n$.

The centre of $\mathbb{H}_{\mathbb{D}}$. Since $\mathbb{H}_{\mathbb{D}}\cong\mathbb{H}\times\mathbb{H}$, the centre is $Z(\mathbb{H})\times Z(\mathbb{H}) \cong \mathbb{R}\times\mathbb{R} \cong \mathbb{D}$; under the identification $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_\mathbb{R}\mathbb{H}$ this is the copy $\mathbb{D}\otimes 1$ spanned by $e_0$ and $j$, as claimed.

The centre of $k[G]$. An element $\sum_g a_g g$ commutes with every $h \in G$ precisely when the coefficient function $a : G \to k$ is constant on conjugacy classes; hence the centre has a basis of class sums and dimension equal to the number of conjugacy classes.

The Units Computed

The unit group is determined in each of the division-algebra-like cases, and the criterion is that the relevant product be invertible in the coefficient ring: over $\mathbb{R}$ and over $\mathbb{C}$ this is the statement that it does not vanish, while for $\mathbb{H}_{\mathbb{D}}$ it takes values in $\mathbb{D}$ and must be a unit there.

Algebra	Unit criterion	Example of a non-unit
$\mathbb{D}$	$(x+y)(x-y) \neq 0$	$1 + j$
$\mathbb{D}'$	$x \neq 0$	$\varepsilon$
$\mathbb{H}$	$q \neq 0$	none
$\mathbb{H}_{\mathbb{D}}$	the relevant value is a unit of $\mathbb{D}$	$1 + j$
$\mathbb{B}$	the relevant value is nonzero	$ie_0 + e_1$

In each case the inverse is the conjugate divided by the relevant product, and the criterion is that the division be legitimate. For $\mathbb{D}$, the elements on the two null lines have no inverse and are zero divisors; for $\mathbb{D}'$, the whole maximal ideal consists of non-units, all nilpotent; for $\mathbb{H}$, there are no non-units besides $0$; for $\mathbb{B}$ the zero divisors are precisely the nonzero elements with vanishing conjugate product. For $\mathbb{H}_{\mathbb{D}}$ the relevant product takes values in $\mathbb{D}$ whose real part $\sum_\mu(q_\mu^2 + q'^2_\mu)$ is positive definite, so it vanishes only at $u = 0$ and does not single out the zero divisors: a nonzero $u$ is a zero divisor exactly when its value is a nonzero zero divisor of $\mathbb{D}$, as for $u = 1+j$, with value $2 + 2j$.

Matrix Algebras

Let $k$ be a field and $n \geq 2$. The algebra $M_n(k)$ of $n\times n$ matrices has dimension $n^2$, its centre is the scalar matrices $kI_n$, and its group of units is the general linear group $\mathrm{GL}_n(k)$. Its two-sided ideals are only $0$ and $M_n(k)$: the algebra is simple. Indeed, if a nonzero ideal contains a matrix $A$ with a nonzero entry $A_{ij}$, then multiplying by the matrix units $E_{ki}$ and $E_{jl}$ produces every matrix unit, hence the whole algebra.

The zero divisors are the nonzero non-invertible matrices, which exist as soon as $n \geq 2$: the matrix units satisfy $E_{11}E_{22} = 0$ with both factors nonzero, and any nonzero matrix of rank less than $n$ kills a nonzero vector on one side. So $M_n(k)$ is simple but not a division algebra, and the two-sided ideal theory is trivial while the one-sided theory is not: the left ideals of $M_n(k)$ are exactly the sets of matrices whose columns lie in a fixed subspace of $k^n$. The matrix algebra is treated in detail.

Polynomial Algebras

Let $k$ be a field and let $k[x_1, \dots, x_n]$ be the polynomial algebra in $n$ commuting indeterminates. It is commutative, infinite-dimensional over $k$, and an integral domain: a product of nonzero polynomials is nonzero. Its group of units is $k^\times$, because only the nonzero constants have inverses; its ideals are finitely generated for every $n$, by the Hilbert basis theorem, and they are principal for $n = 1$, while for $n \geq 2$ they need not be principal. It is not simple, and it has no nonzero nilpotent elements.

For $n = 1$ the structure is completely described by divisibility: every ideal of $k[x]$ is $(f)$ for a unique monic polynomial $f$, the maximal ideals are $(x - a)$ for $a \in k$ when $k$ is algebraically closed, and the units are the nonzero constants. The polynomial algebra is the free commutative $k$-algebra on one generator, and it is the commutative counterpart of the free algebra of Tensor Powers and the Free Algebra; it is treated.

Group Algebras

Let $G$ be a finite group and $k$ a field. The group algebra $k[G]$ has $k$-basis $G$, so $\dim_k k[G] = |G|$, and multiplication is the bilinear extension of the group law; every element of $G$ is a unit, so $G \subseteq k[G]^\times$. The augmentation map $\pi : k[G] \to k$, $\pi\bigl(\sum a_g g\bigr) = \sum a_g$, is an algebra homomorphism, and its kernel is the augmentation ideal $I(G)$, generated by the elements $g - 1_G$; the quotient $k[G]/I(G) \cong k$ is the trivial representation. Over a field of characteristic dividing $|G|$ the algebra is local when $G$ is a $p$-group, with $I(G)$ the unique maximal ideal, and in characteristic zero the structure of $k[G]$ is that of the semisimple decomposition below.

Zero divisors appear as soon as $G$ is nontrivial: if $g \in G$ has order $m \geq 2$, then

$$ (1 - g)(1 + g + \dots + g^{m-1}) = 1 - g^m = 0, $$

with both factors nonzero, so $1 - g$ is a zero divisor. When $\operatorname{char} k$ does not divide $|G|$, Maschke's theorem makes $k[G]$ semisimple, and over $k = \mathbb{C}$ it decomposes as a product of matrix algebras

$$ \mathbb{C}[G] \cong \prod_{\rho \text{ irreducible}} M_{n_\rho}(\mathbb{C}), $$

one factor for each irreducible complex representation, of size equal to its degree; the algebra is simple only for the trivial group. The centre of $k[G]$ consists of the class sums, and it has dimension equal to the number of conjugacy classes. Group algebras are treated, and their operator-algebraic completions.

Summary

The examples fall into a small number of structural types. The division algebras among them are $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$; the fields are the commutative ones, and the simple algebras are those with no nontrivial two-sided ideals, namely $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{B} \cong M_2(\mathbb{C})$ and $M_n(k)$. The algebras with zero divisors or nilpotents — $\mathbb{D}$, $\mathbb{D}'$, $\mathbb{H}_{\mathbb{D}}$, $\mathbb{B}$, $M_n(k)$ for $n\geq2$, $k[G]$ for $|G|\geq2$ — are exactly the ones that are not division algebras, and the two criteria of size of the centre and existence of zero divisors separate the cases: $\mathbb{H}$ is a division algebra with centre $\mathbb{R}$, $\mathbb{B}$ is simple but not a division algebra with centre $\mathbb{C}$, $\mathbb{D}$ is semisimple but not simple with centre $\mathbb{D}$, and $\mathbb{D}'$ is local with a nilpotent radical.

The table at the head of the article records the data for each algebra.

Summary of Notation

Symbol	Meaning
$k$	Ground field
$\mathbb{R}, \mathbb{C}$	Real and complex numbers
$\mathbb{D}$	Split complex numbers, unit $j$, $j^2 = +1$
$\mathbb{D}'$	Dual numbers, unit $\varepsilon$, $\varepsilon^2 = 0$
$\mathbb{H}$	Quaternions, basis $e_0 = 1, e_1, e_2, e_3$
$\mathbb{H}_{\mathbb{D}}$	Split biquaternions, $\mathbb{D}\otimes_\mathbb{R}\mathbb{H}$, $\dim_\mathbb{R} = 8$
$\mathbb{B}$	Biquaternions, $\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$
$M_n(k)$	$n\times n$ matrices over $k$
$k[x_1,\dots,x_n]$	Polynomial algebra
$k[G]$	Group algebra of a finite group
$Z(A)$	Centre of $A$
$A^\times$	Group of units of $A$
$\pi$	Augmentation map $k[G] \to k$