Quaternion Ideals and Simplicity

Introduction

This article treats the ideal structure of the quaternion algebra. Its subject is the statement that $\mathbb{H}$ has no proper one-sided or two-sided ideals, and the consequences of that statement: that $\mathbb{H}$ is simple as a ring, that it is a central simple algebra over its centre, and that its modules over itself are all free. The article is the ideal-theoretic counterpart of Quaternion Norm and Invertibility, which supplies the division property used here as the single input; the two share the statement of division, and the boundary between them is that the quaternion norm article owns the quaternion norm and the unit group, while this article owns the absence of ideals and the central-simplicity statement.

The biquaternion counterpart of this article, Biquaternion Ideals and Peirce Decomposition, develops the same questions in an algebra that is not a division algebra: there the algebra is simple as a ring but carries minimal left ideals and a Peirce decomposition, and it is the direct sum of two such ideals. The contrast is the theme of the last sections of this article, and it is sharp: the properties that the biquaternion algebra has and the quaternion algebra does not are precisely the properties that depend on the presence of idempotents, and $\mathbb{H}$ has no idempotent other than $0$ and $1$.

The corpus's default base is a commutative ring with identity. The quaternion algebra is defined over such a base by its presentation, and the ideal-theoretic statements below hold for an algebra over a commutative ring in which it is a division algebra; the statements that involve the centre and the Brauer group are stated over a field $F$ of characteristic not $2$, and the classical case is $F = \mathbb{R}$. The corpus's convention on the two sibling symbols is respected throughout: $\mathbb{H}_{\mathrm{s}}$ is the split quaternion algebra and $\mathbb{H}_{\mathbb{D}}$ the split biquaternion algebra, and neither is used here except in the comparisons.

Throughout, $\mathbb{H}$ is the quaternion algebra with basis $e_0 = 1, e_1, e_2, e_3$ and $e_k^2 = -e_0$, over a field $F$ of characteristic not $2$ unless stated otherwise; a quaternion is $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with conjugate $\tilde{q}^{\natural} = q_0 e_0 - q_1 e_1 - q_2 e_2 - q_3 e_3$. The quaternion norm $N$, the division property and the unit group $\mathbb{H}^{\times} = \mathbb{H}\setminus\{0\}$ over $\mathbb{R}$ are from Quaternion Norm and Invertibility, named here and not defined again.

Ideals in an Algebra

Definition. Let $A$ be an algebra over a commutative ring $R$. A subset $I\subseteq A$ is a left ideal if it is an additive subgroup with $AI\subseteq I$; a right ideal if $IA\subseteq I$; and a two-sided ideal, or ideal, if it is both. The improper ideals are $0$ and $A$, and an ideal is proper if it is neither.

Proposition. Every left ideal is an $R$-submodule, and the left ideals of $A$ are exactly the submodules of the left $A$-module $A$ itself. If $\tilde q\in A$ then the set $A\tilde q = \{a\tilde q : a\in A\}$ is a left ideal, the principal left ideal generated by $\tilde q$; it is the smallest left ideal containing $\tilde q$.

Proof. The additive subgroup is closed under the $R$-action by the bilinearity of multiplication, so it is a submodule of the left module $A$. Closure of $A\tilde q$ under left multiplication is associativity, and minimality is immediate from closure.

Proposition. For an element $\tilde q$ of an algebra $A$ with a two-sided inverse $\tilde q^{-1}$, the principal left ideal $A\tilde q$ is the whole algebra.

Proof. $1 = \tilde q\tilde q^{-1} = \tilde q(\tilde q^{-1}) \in A\tilde q$, and a left ideal containing $1$ is the whole algebra.

The Ideals of a Division Algebra

Theorem. Let $A$ be an algebra over a commutative ring in which every non-zero element has a two-sided inverse. Then $A$ has no proper left ideal and no proper right ideal.

Proof. Let $I\neq 0$ be a left ideal and let $0\neq \tilde q\in I$. By the hypothesis $\tilde q$ has an inverse, so $1 = \tilde q^{-1}\tilde q \in I$ by closure of the left ideal under left multiplication, and a left ideal containing $1$ is all of $A$. The right case is symmetric.

Corollary. The quaternion algebra $\mathbb{H}$ over $\mathbb{R}$ has no proper left, right or two-sided ideals.

Proof. By Quaternion Norm and Invertibility, every non-zero quaternion is invertible over $\mathbb{R}$, so the theorem applies to $\mathbb{H}$.

The argument uses nothing about the specific multiplication of $\mathbb{H}$; it uses only the division property. This is the sense in which division is the single input of the whole article.

Simplicity of the Quaternion Algebra

Definition. An algebra $A$ is simple if $A^2\neq 0$ and its only two-sided ideals are $0$ and $A$.

Theorem. The quaternion algebra $\mathbb{H}$ is simple over any field over which it is a division algebra; in particular it is simple over $\mathbb{R}$.

Proof. A two-sided ideal is in particular a left ideal, so the absence of proper left ideals is the absence of proper two-sided ideals. The condition $A^2\neq 0$ holds because $1\neq 0$ and $1\cdot1 = 1$.

Remark. The simplicity of $\mathbb{H}$ is inherited by every scalar extension in a precise sense: if $K$ is a field extension of $F$, then $\mathbb{H}\otimes_F K$ is either a division algebra over $K$ and hence simple, or it is isomorphic to the matrix algebra $M_2(K)$, which is simple as a ring as well. In both cases the simplicity survives the extension, although the division property does not.

The Quaternion Algebra as a Central Simple Algebra

Definition. A central simple algebra over a field $F$ is a finite-dimensional simple $F$-algebra whose centre is exactly $F$.

Theorem. The algebra $\mathbb{H}$ is a central simple $F$-algebra of degree two: its centre is $F$, and

$$ \dim_F \mathbb{H} = 4 = 2^2 . $$

Proof. An element commuting with every quaternion is central, and the centre of the quaternion algebra over a field of characteristic not $2$ is the scalar field $F$, as in Quaternion Algebra. Simplicity is the theorem of the preceding section. The dimension is four by the definition of the presentation, and it is the square of the degree $2$.

Definition. The Brauer group $\operatorname{Br}(F)$ of a field $F$ is the group of similarity classes of central simple $F$-algebras under the tensor product, the inverse of a class being the opposite algebra class.

Theorem. The class $[\mathbb{H}]$ of the quaternion algebra in $\operatorname{Br}(F)$ is trivial if and only if $\mathbb{H}$ splits, that is, if and only if $\mathbb{H}\cong M_2(F)$. The algebra is a division algebra exactly when its Brauer class is non-trivial.

Proof. A central simple algebra of degree two is either a division algebra, and then its class is non-trivial, or isomorphic to $M_2(F)$, and then it is a full matrix algebra and its class is the identity of $\operatorname{Br}(F)$. The two cases are distinguished by the existence of zero divisors, by Quaternion Norm and Invertibility.

Remark. Over $F = \mathbb{R}$ the classical quaternion algebra is the only non-split quaternion algebra, so the non-trivial element of $\operatorname{Br}(\mathbb{R})\cong\mathbb{Z}/2\mathbb{Z}$ is exactly $[\mathbb{H}]$. Over $\mathbb{Q}$ there are infinitely many quaternion algebras, classified by a finite set of places of even cardinality.

The Absence of Idempotents and Matrix Units

Definition. An element $e$ of an algebra is idempotent if $e^2 = e$. Two idempotents $e, f$ are orthogonal if $ef = fe = 0$. A set of matrix units is a family $(e_{ij})$ with $e_{ij}e_{kl} = \delta_{jk}e_{il}$.

Theorem. The only idempotents of $\mathbb{H}$ are $0$ and $1$, and $\mathbb{H}$ contains no pair of non-trivial orthogonal idempotents and no matrix units.

Proof. If $e^2 = e$ then $e(e-1) = 0$, and since $\mathbb{H}$ has no zero divisors either $e = 0$ or $e = 1$. Two non-trivial orthogonal idempotents would themselves be zero divisors, since $ef = 0$ with both factors non-zero, so no such pair exists; a matrix-unit family with $n\geq2$ would furnish the non-trivial idempotents $e_{kk}$, and none exists.

This is the point at which the quaternion and biquaternion cases separate. The biquaternion algebra contains the orthogonal idempotents $\tilde\Pi_1 = \tfrac{1}{2}(e_0+ie_3)$ and $\tilde\Pi_2 = \tfrac{1}{2}(e_0-ie_3)$, together with the four matrix units of its decomposition as the sum of the two minimal left ideals $\mathbb{B}\tilde\Pi_1\oplus\mathbb{B}\tilde\Pi_2$; the quaternion algebra contains no such element, because it has no zero divisors, and the absence of the idempotents is the absence of the whole Peirce decomposition.

Minimal Left and Right Ideals

Definition. A non-zero left ideal is minimal if it contains no proper non-zero left ideal.

Theorem. The quaternion algebra has exactly one minimal left ideal, the whole algebra $\mathbb{H}$; equivalently, $\mathbb{H}$ is simple as a left module over itself.

Proof. The only left ideals are $0$ and $\mathbb{H}$. The only non-zero one is $\mathbb{H}$, and it contains no proper non-zero left ideal, so it is minimal; the left module $\mathbb{H}$ has no proper non-zero submodule, so it is simple.

Theorem. Every finitely generated left module over $\mathbb{H}$ is free, of rank equal to its dimension over $F$ divided by four. Consequently the free module of rank $n$ is isomorphic to $\mathbb{H}^n$, and the modules over $\mathbb{H}$ are classified by their rank.

Proof. Over a division ring every module is free, by an argument by induction on a generating set using the invertibility of the coefficients; a finitely generated free module has a well-defined rank because the ring is a division ring. The dimension count is $\dim_F\mathbb{H}^n = 4n$.

The Lattice of Left Ideals

Definition. The lattice of left ideals of an algebra is the partially ordered set of its left ideals under inclusion.

Theorem. The lattice of left ideals of $\mathbb{H}$ is the two-element lattice $\{0,\mathbb{H}\}$, and the same holds for the right ideals.

Proof. There are no proper non-zero left ideals, so the only elements of the lattice are the bounds.

In the biquaternion algebra the corresponding lattice is a projective line, because the minimal left ideals are the lines of a two-dimensional complex vector space and the algebra acts on them as matrices. Here the lattice collapses to its two endpoints, and the collapse is the ideal-theoretic expression of the division property.

The Radical

Definition. The Jacobson radical of an algebra $A$ is the intersection of the maximal left ideals, equivalently the largest two-sided ideal consisting of elements annihilating every simple left module; an algebra is semisimple if its radical is zero.

Theorem. The Jacobson radical of $\mathbb{H}$ is zero, and $\mathbb{H}$ is semisimple and Artinian; as a module over itself it has length one and composition series $0\subsetneq\mathbb{H}$.

Proof. The algebra $\mathbb{H}$ is simple as a left module over itself, since it has no proper non-zero left ideal; it is the only simple left module, and its annihilator is $\{a : a\mathbb{H} = 0\} = 0$ because $1$ acts as the identity. The Jacobson radical, the intersection of the annihilators of the simple left modules, is therefore $0$, and the algebra is semisimple. It is Artinian because it is finite-dimensional over a field, and the chain $0\subsetneq\mathbb{H}$ is a composition series of length one.

The biquaternion algebra is likewise semisimple, and its radical is likewise zero, but its length as a module over itself is two, since it is the direct sum of two minimal left ideals. The length is the numerical expression of the difference between the two cases.

Comparison with the Biquaternion and Split-Biquaternion Cases

The biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ and the split biquaternion algebra $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ carry the same basis as $\mathbb{H}$ with coefficients extended, and their ideal theory differs from that of $\mathbb{H}$ in the two ways in which a semisimple algebra can differ from a division algebra.

Algebra	Centre	Simple	Minimal left ideals	Radically different feature
$\mathbb{H}$	$\mathbb{R}$	yes	one, the algebra itself	division algebra
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	$\mathbb{C}$	yes	two, the columns of $M_2(\mathbb{C})$	complex coefficients
$\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$	$\mathbb{D}$	no	two, one in each summand	$\mathbb{H}_{\mathbb{D}}\cong\mathbb{H}\oplus\mathbb{H}$

The biquaternion algebra is simple over $\mathbb{C}$ but not a division algebra, and it is matrix-like: it has minimal left ideals and a Peirce decomposition, as in Biquaternion Ideals and Peirce Decomposition. The split biquaternion algebra is not simple at all: the idempotents $e_\pm = \tfrac{1}{2}(1\pm j)$ of the split-complex part give a ring decomposition $\mathbb{H}_{\mathbb{D}}\cong\mathbb{H}\oplus\mathbb{H}$, and the two summands are non-trivial two-sided ideals. The quaternion algebra is the case where the idempotents are absent and the two summands degenerate.

The Relation to the Matrix Algebra by Complexification

Theorem. Over the complex field the quaternion algebra becomes a full matrix algebra:

$$ \mathbb{H}\otimes_{\mathbb{R}}\mathbb{C}\cong M_2(\mathbb{C}), $$

an isomorphism of $\mathbb{C}$-algebras.

Proof. The complexification of the quaternion algebra is the quaternion algebra over $\mathbb{C}$, and the complex quaternion algebra is isomorphic to $M_2(\mathbb{C})$ by the standard Pauli-matrix isomorphism, quoted here as standard.

Theorem. The matrix algebra $M_2(\mathbb{C})$ is simple as a ring, but it has exactly two minimal left ideals, the two columns, and every left ideal is a sum of columns; the algebra is the direct sum of its two minimal left ideals.

Proof. Simplicity is the absence of proper two-sided ideals in a full matrix ring; the two column spaces are minimal left ideals, the only left ideals are the four subspaces spanned by subsets of columns, and the sum of the two columns is the whole algebra.

The two theorems together explain the role of the coefficient field. The quaternion algebra is a division algebra over $\mathbb{R}$, with the whole algebra as its only minimal left ideal; after complexification the same algebra becomes the full matrix algebra $M_2(\mathbb{C})$, whose two minimal left ideals appear as soon as the coefficient field is enlarged. The minimal left ideals of the complexification are not the complexification of left ideals of $\mathbb{H}$ — there are none — but they are the complexification of the whole algebra, split by the idempotents of the complexified algebra.

Summary

A left ideal of an algebra that contains an invertible element is the whole algebra. Since every non-zero quaternion is invertible over $\mathbb{R}$ by Quaternion Norm and Invertibility, the quaternion algebra has no proper left, right or two-sided ideal; it is simple as a ring and, being of dimension four with centre $\mathbb{R}$, it is a central simple $\mathbb{R}$-algebra of degree two, with a non-trivial class in the Brauer group and a split complexification.

The absence of proper ideals is the same statement as the absence of idempotents other than $0$ and $1$, and the latter is the absence of zero divisors in another form. Consequently $\mathbb{H}$ has exactly one minimal left ideal, the whole algebra, it is simple as a left module over itself, every finitely generated module over it is free, and its lattice of left ideals collapses to the two endpoints; its Jacobson radical is zero, it is semisimple and Artinian, and its length as a module over itself is one.

The biquaternion algebra shows what is lost in passing away from the division case: it is still simple, but it has minimal left ideals, a lattice of left ideals isomorphic to a projective line, and a length-two decomposition as a module over itself. The split biquaternion algebra loses simplicity as well, decomposing as $\mathbb{H}\oplus\mathbb{H}$. All of these differences are consequences of the presence or absence of idempotents, and hence, ultimately, of the presence or absence of zero divisors.

Summary of Notation

Symbol	Meaning
$\mathbb{H}$	The quaternion algebra, a division algebra over $\mathbb{R}$
$F$	Base field of characteristic not $2$
$e_0 = 1, e_1, e_2, e_3$	Quaternion basis, $e_k^2 = -e_0$
$\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$	General quaternion, conjugate $\tilde{q}^{\natural} = q_0 e_0 - q_1 e_1 - q_2 e_2 - q_3 e_3$
$N$	Quaternion norm, from Quaternion Norm and Invertibility, which supplies the invertibility criterion
$A\tilde q = \{a\tilde q : a\in A\}$	Principal left ideal generated by $\tilde q$
$Z(A)$	Centre of the algebra $A$
$\operatorname{Br}(F)$	Brauer group of the field $F$, $[\mathbb{H}]$ the quaternion class
$M_2(\mathbb{C})$	The full matrix algebra, the complexification of $\mathbb{H}$
$e_\pm = \tfrac{1}{2}(1\pm j)$	Idempotents of $\mathbb{D}$, absent here but present in $\mathbb{H}_{\mathbb{D}}$
$\tilde\Pi_1 = \tfrac{1}{2}(e_0+ie_3)$, $\tilde\Pi_2 = \tfrac{1}{2}(e_0-ie_3)$	Orthogonal idempotents of $\mathbb{B}$, giving its two minimal left ideals
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	Biquaternion algebra, simple but with minimal left ideals
$\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$	Split biquaternion algebra, $\cong\mathbb{H}\oplus\mathbb{H}$, not simple
$\mathbb{H}_{\mathrm{s}}$	Split quaternion algebra, $\cong M_2(\mathbb{R})$, a different system