List of Zero Divisors and Nilpotents

Introduction

This article lists the rings of the corpus that carry zero divisors and those that carry nilpotents — the dual numbers, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z}$, $M_2(\mathbb{R})$, the product rings and the biquaternions — and it separates the three reasons a zero divisor can occur: a nonzero nilpotent, a nontrivial idempotent, and torsion in a finite ring. Every entry points to the article that introduces the object.

The nilpotents are the elements with some power zero, and they form the nilradical when the nilradical is defined; the idempotent zero divisors are the nontrivial idempotents $e$ with $e(1-e) = 0$, and they decompose the ring; and the torsion case is the finite ring $\mathbb{Z}/n\mathbb{Z}$ whose zero divisors come from the factors of $n$ and need not be nilpotent. The three cases are independent, and each ring of the list is placed in exactly one of them where possible.

The article introduces nothing and proves nothing. It records examples and non-examples side by side.

The Three Kinds of Zero Divisor

Kind	The defining relation	A nilpotent?	An idempotent?	Introduced in
nilpotent	$a^n = 0$ for some $n \geq 1$	yes	not in general	Reduced Rings and the Nilradical
idempotent zero divisor	$e^2 = e$ with $e \neq 0, 1$	no	yes	Rings, §§8–9
torsion zero divisor	$n \cdot a = 0$ for some $n \neq 0$	only when $n$ is not squarefree	in general no	Modular Arithmetic and the Ring of Residues

Every nonzero nilpotent is a zero divisor, since $a \cdot a^{n-1} = 0$ with $a^{n-1} \neq 0$ when $n \geq 2$ is the first exponent with $a^n = 0$; a nilpotent is never a nonzero idempotent, because $e^2 = e$ forces $e^n = e$ for all $n$. The reverse implications fail, and the failures are the content of the sections below: $\mathbb{Z}/6\mathbb{Z}$ has zero divisors that are neither nilpotent nor idempotent, and the split-biquaternions have idempotent zero divisors and no nonzero nilpotents.

The Nilpotents

Ring	The nilpotent	Its index	Introduced in
$\mathbb{D}'$, the dual numbers	$\varepsilon$, with $\varepsilon^2 = 0$	$2$	Dual-Numbers Algebra
$\mathbb{Z}/4\mathbb{Z}$	$2$, with $2^2 = 0$	$2$	Reduced Rings and the Nilradical
$\mathbb{Z}/8\mathbb{Z}$	$2$, with $2^3 = 0$	$3$	Modular Arithmetic and the Ring of Residues
$\mathbb{Z}/p^n\mathbb{Z}$	$p$, with $p^n = 0$	$n$	Modular Arithmetic and the Ring of Residues
$k[x]/(x^n)$	$x$, with $x^n = 0$	$n$	Examples of Rings and Fields
$\mathbb{F}_2[C_2]$	$x + 1$, with $(x+1)^2 = 0$	$2$	Examples of Rings and Fields
$M_2(\mathbb{R})$	the strictly upper triangular $N = E_{12}$	$2$	Matrix Algebras
$\mathbb{B}$, the biquaternions	the pure zero divisors satisfying $\tilde{Q}^2 = 0$	$2$	Biquaternion Zero Divisors
$\mathbb{Z}/12\mathbb{Z}$	$6$, with $6^2 = 0$	$2$	Modular Arithmetic and the Ring of Residues

The nilpotents of a commutative ring form an ideal, the nilradical, which is the intersection of the prime ideals; this is developed in Reduced Rings and the Nilradical and Rings, §§6–7. In $\mathbb{B}$ the nilpotents are exactly the pure zero divisors, the elements with vanishing scalar part and $\tilde{Q}^2 = 0$, and they form the nilpotent cone of real dimension $4$ described in Biquaternion Zero Divisors.

The Idempotent Zero Divisors

Ring	The nontrivial idempotents	The decomposition they induce	Introduced in
$\mathbb{D}$, the split-complex numbers	$e_{\pm} = \tfrac{1}{2}(1 \pm j)$	$\mathbb{D} \cong \mathbb{R} \times \mathbb{R}$	Split-Complex Algebra
$\mathbb{Z}/6\mathbb{Z}$	the images of $(1,0)$ and $(0,1)$	$\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$	Modular Arithmetic and the Ring of Residues
$R \times S$	$(1,0)$ and $(0,1)$	the ring is the product by construction	Examples of Rings and Fields
$\mathbb{H}_{\mathbb{D}}$, the split-biquaternions	$\pi_+$, $\pi_-$	$\mathbb{H}_{\mathbb{D}} \cong \mathbb{H} \oplus \mathbb{H}$	Split-Biquaternion Zero Divisors
$M_2(\mathbb{R})$	$E_{11}$, $E_{22}$	the column decomposition $\mathbb{R}^2 = \mathbb{R}e_1 \oplus \mathbb{R}e_2$	Matrix Algebras
a Boolean ring $B$	every element	$B$ is a product of copies of $\mathbb{F}_2$	Von Neumann Regular Rings
$\mathbb{Q}[C_3]$	the orbit sums of the components	$\mathbb{Q}[C_3] \cong \mathbb{Q} \times \mathbb{Q}(\zeta_3)$	Examples of Rings and Fields
$\mathbb{R}[x]/(x^2 - 1)$	the images of the factors	$\mathbb{R}[x]/(x^2-1) \cong \mathbb{R} \times \mathbb{R}$	Examples of Rings and Fields

A nontrivial idempotent $e$ satisfies $e(1-e) = 0$ with $e \neq 0$ and $1 - e \neq 0$, so it is a zero divisor, and it induces the direct sum decomposition $R = Re \oplus R(1-e)$ with the two-sided ideals $Re$ and $R(1-e)$. The decomposition of a ring into a product of two rings is the same thing as a pair of complementary idempotents, and this is the sense in which the idempotents of a commutative Artinian ring give its splitting into local rings, as recorded in List of Reduced, Local and Product Rings.

The Torsion Case

Ring	The zero divisors	Nilpotents?	Idempotents?	Introduced in
$\mathbb{Z}/n\mathbb{Z}$, $n$ squarefree	the nonzero nonunits, all from the factors	none	$2^{\omega(n)}$ of them, one for each factor	Modular Arithmetic and the Ring of Residues
$\mathbb{Z}/6\mathbb{Z}$	$2, 3, 4$	none	the images of $(1,0)$, $(0,1)$	Modular Arithmetic and the Ring of Residues
$\mathbb{Z}/12\mathbb{Z}$	the nonzero nonunits	$6$	from the factors of $12$	Modular Arithmetic and the Ring of Residues
$\mathbb{Z}/p^2\mathbb{Z}$	the multiples of $p$	$p$	none: the ring is local	Modular Arithmetic and the Ring of Residues
$\mathbb{Z}/p^n\mathbb{Z}$	the multiples of $p$	$p$	none: the ring is local	Modular Arithmetic and the Ring of Residues

The torsion case is where the three kinds of zero divisor separate most cleanly: $\mathbb{Z}/6\mathbb{Z}$ has zero divisors and no nilpotents, so it is reduced; $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/p^n\mathbb{Z}$ have nilpotents and no nontrivial idempotents, so they are local and not reduced; and $\mathbb{Z}/12\mathbb{Z}$ has both, since $12$ is neither squarefree nor a prime power. The Chinese remainder theorem records the decomposition, and it is stated in Modular Arithmetic and the Ring of Residues.

The Non-Commutative Cases

Ring	Nilpotents	Idempotent zero divisors	Introduced in
$M_2(\mathbb{R})$	yes: the nilpotent matrices with $N^2 = 0$	yes: $E_{11}$, $E_{22}$	Matrix Algebras
$\mathbb{B}$, the biquaternions	yes: the pure zero divisors	yes: the complex multiples of the idempotents	Biquaternion Zero Divisors
$\mathbb{H}_{\mathbb{D}}$, the split-biquaternions	no nonzero nilpotents	yes: $\pi_+$, $\pi_-$	Split-Biquaternion Zero Divisors
$\mathbb{H}$	none	none	Quaternion Algebra
the upper triangular matrices	yes: the strictly upper triangular matrices	yes: the diagonal idempotents	Matrix Algebras
$k[G]$ for $G$ with torsion	when the characteristic divides the order	from the idempotents of the group algebra	Group Algebras

The two eight-dimensional cases are the contrast the corpus records: $\mathbb{B}$ has both kinds, since the vanishing of its norm produces nilpotents as well as idempotents, while $\mathbb{H}_{\mathbb{D}}$ has only the idempotents, since it is isomorphic to the product $\mathbb{H} \oplus \mathbb{H}$ of two division rings and a product of division rings has no nonzero nilpotents. The matrix ring $M_2(\mathbb{R})$ is semisimple and has no nonzero nilpotent ideal, yet it has nilpotent elements: semisimplicity forbids a nilpotent ideal and not a nilpotent element.

The Reduced Rings

Ring	Why it has no nonzero nilpotent	Introduced in
every domain, for instance $\mathbb{Z}$, $k[x]$, $\mathbb{Z}[i]$	a nilpotent would be a zero divisor	Integral Domains
every field	every nonzero element is a unit	Fields
$\mathbb{Z}/6\mathbb{Z}$	the modulus is squarefree, so no nonzero element has a zero power	Reduced Rings and the Nilradical
a product of reduced rings	the nilpotents of a product are the tuples of nilpotents	Reduced Rings and the Nilradical
a Boolean ring	every element is idempotent, so $a^n = a \neq 0$ for $a \neq 0$	Von Neumann Regular Rings
$\mathbb{H}_{\mathbb{D}}$, the split-biquaternions	isomorphic to $\mathbb{H} \oplus \mathbb{H}$, a product of division rings	Split-Biquaternion Zero Divisors

A ring is reduced when it has no nonzero nilpotent, equivalently when its nilradical is zero. The reduced rings may still have zero divisors, and $\mathbb{Z}/6\mathbb{Z}$ and $\mathbb{H}_{\mathbb{D}}$ are the two standard cases of that phenomenon in the corpus: the first by torsion and the second by idempotents.

Warnings

Object	Why it has no zero divisors and no nilpotents	Introduced in
$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$	integral domains and fields	The Integers, Fields
$k[x]$, $\mathbb{Z}[x]$, $k[x_1, \dots, x_n]$	polynomial rings over a domain	Polynomial Rings and Rational Functions
$\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{-5}]$	quadratic integer rings, hence domains	Examples of Rings and Fields
$\mathbb{H}$	a division ring: every nonzero element is a unit	Quaternion Algebra
$k[F_n]$, the group ring of a free group	torsion-free group, domain	Group Algebras
the zero ring $\{0\}$	$0 = 1$, excluded by the convention $1 \neq 0$	Rings, §2

Summary

This article has listed the rings of the corpus with zero divisors and with nilpotents, separating the three kinds. The nilpotents are $\varepsilon$ in the dual numbers, $2$ in $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/8\mathbb{Z}$, $p$ in $\mathbb{Z}/p^n\mathbb{Z}$, $x$ in $K[x]/(x^n)$, $x+1$ in $\mathbb{F}_2[C_2]$, the nilpotent and strictly upper triangular matrices in $M_2(\mathbb{R})$, and the pure zero divisors of the biquaternions; the idempotent zero divisors are $e_{\pm}$ in the split-complex numbers and the split-biquaternions, the images of $(1,0)$ and $(0,1)$ in $\mathbb{Z}/6\mathbb{Z}$ and in every product ring, the matrix units in $M_2(\mathbb{R})$, the elements of a Boolean ring, and the idempotents of $\mathbb{Q}[C_3]$; and the torsion case separates $\mathbb{Z}/6\mathbb{Z}$, which is reduced, from $\mathbb{Z}/4\mathbb{Z}$, which is not. The reduced rings, the domains and the fields close the list as the objects with no nilpotent and, in the last two cases, no zero divisor at all.

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
$\mathbb{Z}/n\mathbb{Z}$	The residue ring; $\mathbb{Z}/6\mathbb{Z}$ reduced, $\mathbb{Z}/4\mathbb{Z}$ not
$\varepsilon$	The nilpotent generator of the dual numbers, $\varepsilon^2 = 0$
$e_{\pm} = \tfrac{1}{2}(1 \pm j)$	The idempotents of the split-complex numbers and the split-biquaternions
$E_{11}$, $E_{22}$, $E_{12}$	Matrix units; $E_{12}$ is nilpotent
$\mathbb{D}$, $\mathbb{D}'$	Split-complex numbers, dual numbers
$\mathbb{B}$, $\mathbb{H}_{\mathbb{D}}$	Biquaternions with nilpotents, split-biquaternions without
$\mathbb{H}$	The quaternions, with neither
$\tilde{Q}$	A biquaternion
$B$	A Boolean ring
$[a]$	The class of $a$ in a residue ring