Zero Divisors of the Split-Complex Algebra

Introduction

This article studies the zero divisors of the split-complex algebra $\mathbb{D}$. It follows Norm and Invertibility in the General Split-Complex Algebra, which established the criterion of invertibility and the three-way classification of the elements, and it opens the element theory of the category: the two families it separates are developed in Square Roots of One and the Idempotents of the Split-Complex Algebra and Square Roots of Zero and the Nilpotents of the Split-Complex Algebra, and its vocabulary of the two lines is used throughout. The goal here is to characterize the zero divisors of the general plain bilinear product, to split them into two families, and to describe their structure.

The results. A nonzero split-complex number is a zero divisor of the plain product exactly when one of its idempotent coordinates vanishes, equivalently when its norm vanishes, $Q\bar Q=q^{2}-q'^{2}=0$; the elements therefore fall into the three classes invertible, zero and zero divisor. The zero divisors split into two disjoint families, generated by the primitive zero divisors $\Pi_{+}$ and $\Pi_{-}$: the nonzero real multiples of $\Pi_{+}$ and of $\Pi_{-}$, lying on the two null lines $\mathbb{R}\Pi_{+}$ and $\mathbb{R}\Pi_{-}$; and their two sign classes on each line, forming four connected components. The zero-divisor set is a cross of two lines minus the origin, of real dimension $1$; it is not an ideal; its closure $\mathcal{N}$ is closed under multiplication; and each line is the annihilator of the other.

A divisor is relative to a product. The definition is stated once for a general product and then read on the products of the space; the unqualified "zero divisor" means "zero divisor of the general plain bilinear product", the multiplication of the split-complex algebra, and every other product is named. The two products and their comparison are The Two Products of the Split-Complexoid Space and Relations and Comparison Between the Two Products of the Split-Complexoid Space.

Every claim is proved or stated as a definition, no physics is invoked and no examples are given. The algebra $\mathbb{D}=\mathbb{R}[j]$, $j^{2}=+1$, its conjugation and its idempotents are assumed from Split-Complex Algebra and Square Roots of One and the Idempotents of the Split-Complex Algebra, and the criterion of invertibility from Norm and Invertibility in the General Split-Complex Algebra. The annihilators are Annihilators of the Split-Complex Algebra, and the ideals are Ideals and Peirce Decomposition of the Split-Complex Algebra.

Throughout this article, a general element is $Q=q+jq'$ with $q,q'\in\mathbb{R}$, the conjugate is $\bar Q=q-jq'$, and the idempotents are $\Pi_{\pm}=\tfrac12(1\pm j)$, with idempotent coordinates $Q=Q_{+}\Pi_{+}+Q_{-}\Pi_{-}$ and $Q_{\pm}=q\pm q'$. The null cone is

$$ \mathcal{N}=\{Q:Q\bar Q=0\}=\{Q:q^{2}=q'^{2}\}=\mathbb{R}\Pi_{+}\cup\mathbb{R}\Pi_{-}, $$

and the zero-divisor set is $\mathcal{Z}=\mathcal{N}\setminus\{0\}$. The norm is $\langle Q\rangle_{*}=Q\bar Q=q^{2}-q'^{2}$; its vanishing is the algebraic condition that decides the zero divisors of the plain product, and it is never read here as a length. The other algebraic norm, $\langle Q\rangle=Q^{2}$, does not decide them (Norm and Invertibility in the Split-Complex Algebra).

Definition

The word names a product, so the definition is stated for a general product and then read on the products of this space. Let $\star$ be a product on $\mathbb{D}$.

Definition. Let $\star$ be a product on $\mathbb{D}$ and let $Q_1\in\mathbb{D}$. The element $Q_1$ is a left zero divisor of $\star$ when $Q_1\neq0$ and there is a nonzero $Q_2$ with $Q_1\star Q_2=0$; it is a right zero divisor of $\star$ when $Q_1\neq0$ and there is a nonzero $Q_2$ with $Q_2\star Q_1=0$; and it is a two-sided zero divisor of $\star$ when it is both. The left annihilator and the right annihilator of $Q_1$ are

$$ A_\star(Q_1)=\{Q_2:Q_1\star Q_2=0\},\qquad B_\star(Q_1)=\{Q_2:Q_2\star Q_1=0\}. $$

The general definition is Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §The Zero Divisor and the Annihilator, read on this two-dimensional member of the family. When $\star$ is the general plain bilinear product, the multiplication of the algebra, the adverb "plain" is used and the product is written by juxtaposition; unqualified, zero divisor means zero divisor of the plain product, and annihilator means the annihilator of the plain product.

Proposition. The plain product is commutative, so for every $Q_1$ the left and the right zero-divisor conditions coincide, the two annihilators coincide, and the three adjectives collapse to one: an element is a zero divisor of the plain product exactly when it is a left zero divisor, a right zero divisor, or a two-sided zero divisor of it.

Proof. The plain product is commutative, so $Q_1Q_2=Q_2Q_1$ and the two equations of the definition are the same equation.

Remark (the zero element). The zero element is excluded from the definition. It annihilates every element, so its annihilator is the whole algebra and not a proper ideal; it is not a zero divisor, and the zero-divisor set is the null cone with the origin removed.

The Criterion

A nonzero split-complex number is a zero divisor of the plain product exactly when one of its idempotent coordinates vanishes, equivalently exactly when $q^{2}=q'^{2}$, equivalently exactly when $Q\bar Q=0$.

Theorem (the criterion). For $Q\in\mathbb{D}$ with $Q\neq0$ the following are equivalent:

1. $Q$ is a zero divisor of the plain product;
2. $Q_{+}Q_{-}=0$;
3. $q^{2}=q'^{2}$;
4. $Q\bar Q=0$.

Proof. In the idempotent coordinates, $Q=Q_{+}\Pi_{+}+Q_{-}\Pi_{-}$ and $B=B_{+}\Pi_{+}+B_{-}\Pi_{-}$, and the product is componentwise,

$$ QB=Q_{+}B_{+}\Pi_{+}+Q_{-}B_{-}\Pi_{-}. $$

So $QB=0$ exactly when $Q_{+}B_{+}=0$ and $Q_{-}B_{-}=0$. If $Q$ is a zero divisor, choose $B\neq0$; at least one of $B_{+},B_{-}$ is nonzero. If $B_{+}\neq0$ then $Q_{+}=0$, and if $B_{-}\neq0$ then $Q_{-}=0$; in either case $Q_{+}Q_{-}=0$. Conversely, if $Q_{+}Q_{-}=0$ and $Q\neq0$, one coordinate vanishes and the other does not; if $Q_{+}=0$ then $Q=Q_{-}\Pi_{-}\neq0$ and $Q\Pi_{+}=0$ with $\Pi_{+}\neq0$, so $Q$ is a zero divisor, and the case $Q_{-}=0$ is symmetric. The equivalence with $q^{2}=q'^{2}$ is $Q_{+}Q_{-}=(q+q')(q-q')=q^{2}-q'^{2}$, and the equivalence with $Q\bar Q=0$ is the definition of the norm.

Corollary. The zero-divisor set is exactly the nonzero part of the null cone,

$$ \boxed{\;\mathcal{Z}=\{Q\neq0:q^{2}=q'^{2}\}=\{Q:Q_{+}=0\text{ or }Q_{-}=0,\ Q\neq0\}\;}. $$

Remark (no norm of the plain product is used). The criterion is expressed by the idempotent coordinates and by the norm $\langle Q\rangle_{*}=Q\bar Q$; it is not expressed by the algebraic norm of the plain product, $\langle Q\rangle=Q^{2}$, whose vanishing set is the origin (Norm and Invertibility in the Split-Complex Algebra). The two algebraic norms are different objects, and the zero divisors are the vanishing set of the second.

The Three-Way Classification

The elements of $\mathbb{D}$ fall into exactly three classes under the plain product: the invertible elements, the zero element, and the zero divisors.

Theorem. For $Q\in\mathbb{D}$ exactly one of the following holds:

1. $Q$ is invertible;
2. $Q=0$;
3. $Q$ is a zero divisor.

Proof. If $Q\neq0$ and $Q$ is not a zero divisor, then the plain equation $QB=1$ has a solution by the criterion of Norm and Invertibility in the General Split-Complex Algebra, so $Q$ is invertible. If $Q\neq0$ and $Q$ is a zero divisor, it is in the third class. The three classes are disjoint by their definitions, and the zero element is neither invertible nor a zero divisor.

Corollary (the non-units). The non-units of $\mathbb{D}$ are the zero element and the zero divisors. They do not form an ideal: $\Pi_{+}+\Pi_{-}=1$ is a unit, so the sum of two non-units can be a unit. The algebra is therefore not a local ring, and it has no unique maximal ideal of non-units.

The Algebra Is Not a Division Algebra

The plain product has nonzero zero divisors, so the split-complex algebra is not a division algebra and not a field.

The identity

$$ (1+j)(1-j)=1-j^{2}=0 $$

exhibits two nonzero elements with zero product. The algebra is commutative, associative and unital, but it is not a domain and not a field; the zero divisors are exactly the obstructions.

Proposition. The plain product is associative, commutative, unital and reduced: it has no nonzero nilpotent element. It has zero divisors, so it is not a domain; it is a product of two copies of $\mathbb{R}$, and it is not a field.

Proof. The algebra is $\mathbb{R}[j]/(j^{2}-1)\cong\mathbb{R}\times\mathbb{R}$ by $Q\mapsto(Q_{+},Q_{-})$, a product of two fields; a product of two fields has the idempotents $(1,0)$ and $(0,1)$, hence zero divisors, and has no nonzero nilpotent because a nilpotent has both coordinates nilpotent and $\mathbb{R}$ is reduced. The field statement follows because a field has no zero divisors.

The zero divisors are therefore not a defect of a failed inversion formula but the visible form of the decomposition $\mathbb{D}\cong\mathbb{R}\oplus\mathbb{R}$. The two components are the two minimal ideals, and a zero divisor is an element supported on exactly one of them.

The Two Families of Zero Divisors

The zero-divisor set splits into the two disjoint families $\mathbb{R}^{\times}\Pi_{+}$ and $\mathbb{R}^{\times}\Pi_{-}$, the two null lines with the origin removed; each family is a line through the origin and each is the annihilator of the other.

The factorisation $q^{2}-q'^{2}=(q-q')(q+q')$ gives

$$ \mathcal{N}=\{q=q'\}\cup\{q=-q'\}=\mathbb{R}(1+j)\cup\mathbb{R}(1-j)=\mathbb{R}\Pi_{+}\cup\mathbb{R}\Pi_{-}. $$

The two lines meet only at the origin, because $\Pi_{+}$ and $\Pi_{-}$ are linearly independent. Removing the origin,

$$ \mathcal{Z}=\mathbb{R}^{\times}\Pi_{+}\;\cup\;\mathbb{R}^{\times}\Pi_{-}, $$

a disjoint union of the two families

$$ \mathcal{Z}_{+}=\mathbb{R}^{\times}\Pi_{+}=\{(t,t):t\neq0\},\qquad \mathcal{Z}_{-}=\mathbb{R}^{\times}\Pi_{-}=\{(t,-t):t\neq0\}, $$

where $Q=(q,q')$ in the basis $1,j$.

Proposition (the primitive zero divisors). Every zero divisor is a nonzero real multiple of exactly one of the two idempotents. The idempotents $\Pi_{+}$ and $\Pi_{-}$ are the primitive zero divisors, and

$$ \boxed{\;\mathcal{Z}=\mathbb{R}^{\times}\Pi_{+}\cup\mathbb{R}^{\times}\Pi_{-}\;}. $$

Proof. A point of $\mathbb{R}^{\times}\Pi_{+}$ is $t\Pi_{+}$ with $t\neq0$, and a point of $\mathbb{R}^{\times}\Pi_{-}$ is $s\Pi_{-}$ with $s\neq0$; the two lines meet only at $0$, so the multiple and the line are unique.

Proposition (the annihilators). The annihilator of a nonzero multiple of $\Pi_{+}$ is the opposite line, and symmetrically:

$$ \boxed{\;\operatorname{ann}(\lambda\Pi_{+})=\mathbb{R}\Pi_{-},\qquad \operatorname{ann}(\lambda\Pi_{-})=\mathbb{R}\Pi_{+},\qquad \lambda\neq0\;}. $$

Proof. Write $Q=\lambda\Pi_{+}$ and $B=B_{+}\Pi_{+}+B_{-}\Pi_{-}$. Then $QB=\lambda B_{+}\Pi_{+}$, which vanishes exactly when $B_{+}=0$, that is $B\in\mathbb{R}\Pi_{-}$. The other case is symmetric.

So the two families annihilate one another: each null line is the annihilator of the other. The annihilators are computed in full in Annihilators of the Split-Complex Algebra.

The Structure of the Zero-Divisor Set

The zero-divisor set is the union of two lines through the origin with the origin removed; it has real dimension $1$, four connected components, and its closure $\mathcal{N}$ is closed under multiplication, while $\mathcal{Z}$ itself is not; neither is closed under addition.

Dimension and Components

Each family $\mathbb{R}^{\times}\Pi_{\pm}$ is a line with the origin removed. The line $\mathbb{R}^{\times}\Pi_{+}$ has two connected components, the positive ray $\{t\Pi_{+}:t>0\}$ and the negative ray $\{t\Pi_{+}:t<0\}$, and the same for $\Pi_{-}$. Hence $\mathcal{Z}$ has four connected components, although its closure $\mathcal{N}$ is connected: the two lines cross at the origin, which is not in $\mathcal{Z}$. The real dimension is $1$ at every point.

Closure under Multiplication, Not under Addition

Proposition. The set $\mathcal{Z}\cup\{0\}=\mathcal{N}$ is closed under multiplication. The set $\mathcal{Z}$ is not closed under multiplication: the product of two elements of the same family lies in $\mathcal{Z}$, but the product of one element from each family is $0$. Neither set is closed under addition, and neither is an ideal of $\mathbb{D}$.

Proof. For the product, $\Pi_{+}^{2}=\Pi_{+}$, $\Pi_{-}^{2}=\Pi_{-}$ and $\Pi_{+}\Pi_{-}=0$, so the product of two elements of the same line stays on that line and the product of two elements of opposite lines is zero, outside $\mathcal{Z}$ but in $\mathcal{N}$. For the sum, $\Pi_{+}+\Pi_{-}=1$ is a unit and not in $\mathcal{N}$; so the sum of two zero divisors can be a unit, and $\mathcal{N}$ is not closed under addition. An ideal is closed under addition, so $\mathcal{N}$ is not an ideal; the two lines individually are ideals, and they are the minimal ideals of the algebra.

Remark (the zero-divisor set is not a submanifold at the origin). The set $\mathcal{N}$ is a cross, not a submanifold at $0$; removing the origin makes $\mathcal{Z}$ a disjoint union of four open rays, each a one-dimensional submanifold. The closure adds the singular point.

The Two Sign Classes on Each Family

On $\mathcal{Z}_{+}$ the parameter $t$ has a sign, and the two signs are the two components of the family; the same on $\mathcal{Z}_{-}$. The four components are

$$ \{t\Pi_{+}:t>0\},\quad \{t\Pi_{+}:t<0\},\quad \{t\Pi_{-}:t>0\},\quad \{t\Pi_{-}:t<0\}. $$

They are also the four sign classes of the pair of idempotent coordinates on the zero-divisor set: on $\mathcal{Z}_{+}$ one has $Q_{+}=t$ and $Q_{-}=0$, so the sign of $Q_{+}$ separates the two rays; on $\mathcal{Z}_{-}$ the sign of $Q_{-}$ does the same. Off the null cone the pair of signs $(\operatorname{sgn}Q_{+},\operatorname{sgn}Q_{-})$ separates the complement into four sign classes, and each zero-divisor ray is a boundary between two of them.

The Zero-Divisor Set and the Distinguished Subspaces

The two eigenspaces of the conjugation contain no zero divisor except the origin; the two idempotent lines contain only zero divisors besides the origin; the null cone meets the eigenspaces only at the origin.

The distinguished one-dimensional subspaces are the real line $\mathbb{R}_{\mathbb{D}}$, the split imaginary line $j\mathbb{R}_{\mathbb{D}}$, and the two idempotent lines $\mathbb{R}\Pi_{+}$ and $\mathbb{R}\Pi_{-}$.

The Real Subspace $\mathbb{R}_{\mathbb{D}}$

On $\mathbb{R}_{\mathbb{D}}$ one has $Q=q$, $Q_{+}=Q_{-}=q$, and $Q\bar Q=q^{2}$, which vanishes only at $q=0$. Every nonzero real number is a unit; there is no zero divisor.

The Split Imaginary Subspace $j\mathbb{R}_{\mathbb{D}}$

On $j\mathbb{R}_{\mathbb{D}}$ one has $Q=jq'$, $Q_{+}=q'$, $Q_{-}=-q'$, and $Q\bar Q=-q'^{2}$, which vanishes only at $q'=0$. Every nonzero element is a unit; there is no zero divisor.

The Idempotent Lines

On $\mathbb{R}\Pi_{+}$ one has $Q=t\Pi_{+}$, so $Q_{+}=t$ and $Q_{-}=0$; the norm vanishes identically, and every nonzero point is a zero divisor, with annihilator $\mathbb{R}\Pi_{-}$. On $\mathbb{R}\Pi_{-}$ the roles are exchanged: every nonzero point is a zero divisor, with annihilator $\mathbb{R}\Pi_{+}$. The two lines are the two minimal ideals.

Summary of the Distribution

So the zero divisors are exactly the nonzero points of the two idempotent lines, and the two eigenspaces of the conjugation are totally free of them away from the origin.

The Zero Divisors and the Idempotents

The zero divisors are exactly the elements supported on a single idempotent; the primitive zero divisors are the two idempotents themselves, and every other zero divisor is a nonzero real multiple of one of them.

In the idempotent basis,

$$ Q=Q_{+}\Pi_{+}+Q_{-}\Pi_{-}, $$

so $Q$ is a zero divisor exactly when one coordinate vanishes and the other does not:

$$ \mathcal{Z}=\{Q:Q_{+}=0,\ Q_{-}\neq0\}\cup\{Q:Q_{-}=0,\ Q_{+}\neq0\}. $$

The two families are the two coordinate lines $\mathbb{R}\Pi_{\pm}$ with the origin removed. The idempotents $\Pi_{\pm}$ are the primitive zero divisors, and every other zero divisor is a real multiple of one of them. The conjugation exchanges the two families, $\overline{\Pi_{+}}=\Pi_{-}$, and preserves each norm, so it maps a zero divisor to a zero divisor on the opposite line.

The zero divisors are also the elements of rank one in the regular matrix model: the matrix of $Q$ is $\begin{pmatrix}q&q'\\q'&q\end{pmatrix}$, of determinant $q^{2}-q'^{2}$ and trace $2q$; it is singular exactly on the null cone, and on the nonzero part of the cone it has rank one. The zero matrix is the origin. The matrix model is The Matrix Element Representations of the Split-Complexoid.

The Zero Divisors and the Two Products

The zero divisors are a property of the plain product; the general plain sesquilinear product $Q_1\bar Q_2$ has the same zero-divisor set, and the comparison of the two products preserves the set while changing the annihilator.

The plain product and the sesquilinear product agree on the zero-divisor set: $Q_1Q_2=0$ for some nonzero $Q_2$ exactly when $Q_1\bar Q_2=0$ for some nonzero $Q_2$, because the conjugation is a bijection and $\bar Q_2$ ranges over the nonzero elements. The annihilators, however, are not the same: on the null line $\mathbb{R}\Pi_{+}$ the plain annihilator is the opposite line $\mathbb{R}\Pi_{-}$, while the sesquilinear annihilator is the line $\mathbb{R}\Pi_{+}$ itself, since $\Pi_{+}\bar\Pi_{+}=\Pi_{+}\Pi_{-}=0$. The comparison of the two products, their associativity and their one-sided units are Relations and Comparison Between the Two Products of the Split-Complexoid Space; the zero-divisor set of the four general products of the larger family is Zero Divisors in the Algebraic Structures over the Quaternioid $\mathbb{R}$ Space.

Summary

A nonzero split-complex number $Q$ is a zero divisor of the plain product exactly when one of its idempotent coordinates vanishes, equivalently when $q^{2}=q'^{2}$, equivalently when $Q\bar Q=0$. The zero-divisor set is the null cone with the origin removed,

$$ \mathcal{Z}=\mathbb{R}^{\times}\Pi_{+}\cup\mathbb{R}^{\times}\Pi_{-}, $$

the union of the two null lines, of real dimension $1$ and with four connected components. The two families annihilate one another, $\operatorname{ann}(\lambda\Pi_{+})=\mathbb{R}\Pi_{-}$ and $\operatorname{ann}(\lambda\Pi_{-})=\mathbb{R}\Pi_{+}$, and the primitive zero divisors are the two idempotents $\Pi_{\pm}$.

The elements fall into the three classes invertible, zero and zero divisor; the non-units are the zero element and the zero divisors and do not form an ideal, since $\Pi_{+}+\Pi_{-}=1$. The closure $\mathcal{N}$ is closed under multiplication, while the zero-divisor set itself is not, because the product of the two families is $0$; neither is closed under addition, neither is an ideal, and the closure is the singular cross $\mathcal{N}$. On the two eigenspaces of the conjugation the norm is definite and there are no zero divisors except the origin, while the two idempotent lines are totally singular; the conjugation exchanges the two lines. The zero divisors are the elements supported on a single idempotent, the rank-one elements of the regular matrix model, and the common zero-divisor set of the two products of the split-complexoid.

Summary of Notation

Further Reading

• Norm and Invertibility in the General Split-Complex Algebra, for the criterion of invertibility and the three-way classification.
• Annihilators of the Split-Complex Algebra, for the annihilator of a zero divisor and the multiplication operators.
• Square Roots of One and the Idempotents of the Split-Complex Algebra, for the two idempotents and their arithmetic.
• Square Roots of Zero and the Nilpotents of the Split-Complex Algebra, for the absence of nonzero nilpotents in the algebra.
• Ideals and Peirce Decomposition of the Split-Complex Algebra, for the two minimal ideals and the Peirce decomposition.
• The Two Products of the Split-Complexoid Space, for the two products and their comparison.