List of Rings by Their Idempotents
Introduction
This article lists the rings of the corpus by their idempotents — the elements $e$ with $e^2 = e$ — the direct sum decompositions those idempotents induce, the Peirce decomposition of a module or an algebra relative to an orthogonal family of them, and the two extreme cases: the von Neumann regular rings, in which every ideal is idempotent-generated, and the domains and local rings, in which the only idempotents are $0$ and $1$. Every entry points to the article that introduces the object.
The idempotents of a ring are indifferent to its additive and ideal structure and sensitive to its decomposition: a nontrivial idempotent $e$ with $e \neq 0, 1$ splits the ring as $R = Re \oplus R(1-e)$ and its module as $M = eM \oplus (1-e)M$, while a ring with no nontrivial idempotent is indecomposable and is called connected. The list records each ring with its idempotents, with the decomposition they give, and with the corner algebra $eRe$ that the Peirce decomposition isolates.
The article introduces nothing and proves nothing. It records examples and non-examples side by side.
The Idempotents of Each Ring
| Ring | The idempotents | Their number | Introduced in |
|---|---|---|---|
| a field $k$ | $0, 1$ | $2$ | Fields |
| a domain $R$ | $0, 1$ | $2$ | Integral Domains |
| a local ring | $0, 1$ | $2$ | Localization and the Fraction Field |
| $\mathbb{H}$, a division ring | $0, 1$ | $2$ | Quaternion Algebra |
| $\mathbb{Z}/4\mathbb{Z}$ | $0, 1$ | $2$ | Reduced Rings and the Nilradical |
| $\mathbb{Z}/p^n\mathbb{Z}$ | $0, 1$ | $2$ | Reduced Rings and the Nilradical |
| $\mathbb{D}' = \mathbb{R}[\varepsilon]/(\varepsilon^2)$ | $0, 1$ | $2$ | Dual-Numbers Algebra |
| $\mathbb{F}_2[C_2] \cong \mathbb{F}_2[x]/(x+1)^2$ | $0, 1$ | $2$ | Examples of Rings and Fields |
| $\mathbb{Z}/6\mathbb{Z}$ | $0, 1, 3, 4$ | $4$ | Modular Arithmetic and the Ring of Residues |
| $\mathbb{Z}/12\mathbb{Z}$ | $0, 1, 4, 9$ | $4$ | Modular Arithmetic and the Ring of Residues |
| $\mathbb{Z}/n\mathbb{Z}$ | one for each factorisation into coprime parts | $2^{\omega(n)}$ | Modular Arithmetic and the Ring of Residues |
| $\mathbb{D} = \mathbb{R}[j]/(j^2-1)$ | $0, 1, \pi_+, \pi_-$ | $4$ | Split-Complex Algebra |
| $\mathbb{R}[x]/(x^2-1)$ | $0, 1$, the two class idempotents | $4$ | Examples of Rings and Fields |
| $\mathbb{H}_{\mathbb{D}}$, the split-biquaternions | $0, \pi_+, \pi_-, e_0$ | $4$ | Split-Biquaternion Algebra |
| $\mathbb{Q}[C_3]$ | $0, 1$ and the two minimal ones | $4$ | Examples of Rings and Fields |
| $M_2(\mathbb{R})$ | the projections | infinitely many | Matrix Algebras |
| $\mathbb{B}$, the biquaternions | the complex multiples of the idempotents | infinitely many | Biquaternion Ideals and Peirce Decomposition |
| a Boolean ring $B$ | every element | $\lvert B \rvert$ | Von Neumann Regular Rings |
The dichotomy the table records is between the rings that are connected, with $0$ and $1$ as their only idempotents, and the rings that split. The domains, the fields, the division rings and the local rings are connected; $\mathbb{Z}/n\mathbb{Z}$ has $2^{\omega(n)}$ idempotents, where $\omega(n)$ counts the distinct primes dividing $n$, so $\mathbb{Z}/6\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$ have four; and the Boolean rings have an idempotent for every element. The split-biquaternions and the split-complex numbers have exactly four idempotents each, and $\mathbb{H}$ has two, which is the sharpest contrast between the quaternion family and its split relatives.
The Direct Sum Decompositions
| Ring | The idempotent $e$ | The decomposition | Introduced in |
|---|---|---|---|
| any ring | $e$ | $R = Re \oplus R(1-e)$ | Rings, §§8–9 |
| $\mathbb{Z}/6\mathbb{Z}$ | $3$ | $\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ | Modular Arithmetic and the Ring of Residues |
| $\mathbb{D}$ | $\pi_+$ | $\mathbb{D} \cong \mathbb{R} \pi_+ \oplus \mathbb{R} \pi_- \cong \mathbb{R} \times \mathbb{R}$ | Split-Complex Algebra |
| $\mathbb{H}_{\mathbb{D}}$ | $\pi_+$ | $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\pi_+ \oplus \mathbb{H}\pi_-$ | Split-Biquaternion Zero Divisors |
| $R \times S$ | $(1,0)$ | the product by construction | Examples of Rings and Fields |
| $\mathbb{Q}[C_3]$ | the minimal idempotents | $\mathbb{Q}[C_3] \cong \mathbb{Q} \times \mathbb{Q}(\zeta_3)$ | Examples of Rings and Fields |
| $\mathbb{R}[x]/(x^2-1)$ | $e_{\pm}$ | $\mathbb{R}[x]/(x^2-1) \cong \mathbb{R} \times \mathbb{R}$ | Examples of Rings and Fields |
| a Boolean ring $B$ | every idempotent $e$ | $B = Be \oplus B(1-e)$ for every $e$, every element being idempotent | Von Neumann Regular Rings |
| a commutative Artinian ring | the primitive idempotents | the product of the local rings $R e_i$ | Noetherian and Artinian Rings |
A nontrivial idempotent $e$ satisfies $e(1-e) = 0$, so $R = Re \oplus R(1-e)$ is a direct sum of the two-sided ideals $Re$ and $R(1-e)$, and the two ideals are rings in their own right with identities $e$ and $1-e$. A family of idempotents that are orthogonal, $e_i e_j = 0$ for $i \neq j$, and complete, $\sum e_i = 1$, gives the decomposition $R = \bigoplus_i R e_i$; the primitive idempotents are those that cannot be split further, and for a commutative Artinian ring the primitive decomposition is the product of its localisations, as recorded in Noetherian and Artinian Rings and in List of Reduced, Local and Product Rings.
The Peirce Decomposition
| Object | The orthogonal family | The decomposition | Introduced in |
|---|---|---|---|
| a module $M$ over $R$ | a single idempotent $e$ | $M = eM \oplus (1-e)M$ | Modules |
| an algebra $A$ | a single idempotent $e$ | $A = eAe \oplus eA(1-e) \oplus (1-e)Ae \oplus (1-e)A(1-e)$ | Biquaternion Ideals and Peirce Decomposition |
| $\mathbb{B} \cong M_2(\mathbb{C})$ | $p = E_{11}$, $q = E_{22}$ | the four one-dimensional corners | Biquaternion Ideals and Peirce Decomposition |
| $M_2(\mathbb{R})$ | $E_{11}, E_{22}$ | the column decomposition of $\mathbb{R}^2$ | Matrix Algebras |
| $R \times S$ | $(1,0)$, $(0,1)$ | the two components | Examples of Rings and Fields |
| a group algebra $k[G]$ | the idempotents of the group | $k[G] = \bigoplus e_i k[G] e_i \oplus \text{off-diagonal}$ | Group Algebras |
The Peirce decomposition relative to a complete orthogonal family of idempotents $e_1 + \cdots + e_n = 1$ writes a module as $M = \bigoplus_i e_i M$ and an algebra as the direct sum of the corner algebras $e_i A e_j$. The biquaternion case is the corpus's worked example: with $p = E_{11}$ and $q = E_{22}$ the four corners are one-dimensional over $\mathbb{C}$, and the off-diagonal ones are spanned by the nilpotents $x$ and $y$, as recorded in Biquaternion Ideals and Peirce Decomposition.
The von Neumann Regular Rings
| Ring | Why every ideal is idempotent-generated | Introduced in |
|---|---|---|
| a Boolean ring | every element is idempotent, so every ideal is generated by idempotents | Von Neumann Regular Rings |
| a product of fields | the primitive idempotents generate the principal ideals | Examples of Rings and Fields |
| $\prod_i k_i$ over a family of fields | the idempotents of the product generate the ideals | Examples of Rings and Fields |
| a commutative von Neumann regular ring | each localisation at a maximal ideal is a field, and every principal ideal is generated by an idempotent | Von Neumann Regular Rings |
| a reduced ring of Krull dimension zero | equivalently von Neumann regular | Von Neumann Regular Rings |
| $\mathbb{Z}$, $k[x]$, $\mathbb{Z}[\sqrt{-5}]$ | not von Neumann regular: the ideal $(2)$ is not generated by an idempotent | Integral Domains |
| $\mathbb{Z}/4\mathbb{Z}$ | not von Neumann regular: $2 \cdot 2 = 0$, and the maximal ideal is not idempotent-generated | Reduced Rings and the Nilradical |
A ring is von Neumann regular when every principal left ideal is generated by an idempotent, equivalently when every finitely generated left ideal is idempotent-generated. The commutative von Neumann regular rings are exactly the reduced rings of Krull dimension zero, equivalently those whose localisation at every maximal ideal is a field; the Boolean rings are the characteristic-two examples. The domains and the local rings are the non-examples: in a domain the principal ideal $(a)$ for $a$ neither zero nor a unit is not generated by an idempotent, since the only idempotents are $0$ and $1$, and $\mathbb{Z}/4\mathbb{Z}$ fails because $2 \cdot 2 = 0$. The class is the subject of Von Neumann Regular Rings.
The Rings with Few Idempotents
| Ring | The idempotents | Why | Introduced in |
|---|---|---|---|
| a domain | $0, 1$ | $e(1-e) = 0$ forces $e = 0$ or $1$ | Integral Domains |
| a local ring | $0, 1$ | the nonunits form the maximal ideal, and a nontrivial idempotent is a nonunit with a nonunit complement | Localization and the Fraction Field |
| a division ring | $0, 1$ | a nonzero idempotent is a unit, and the only unit idempotent is $1$ | Division Rings |
| $\mathbb{Z}/p^n\mathbb{Z}$ | $0, 1$ | a local ring | Reduced Rings and the Nilradical |
| $\mathbb{D}'$ | $0, 1$ | a local ring | Dual-Numbers Algebra |
| $\mathbb{F}_2[C_2]$ | $0, 1$ | isomorphic to $\mathbb{F}_2[x]/(x+1)^2$, a local ring | Examples of Rings and Fields |
| a connected ring | $0, 1$ | the definition of connected | Rings, §§8–9 |
A ring with $0$ and $1$ as its only idempotents is connected, equivalently indecomposable: it is not the product of two nonzero rings. Every domain, every local ring and every division ring is connected, and a nontrivial idempotent is exactly what a product fails to exclude. The scarcity of idempotents is therefore not a defect but the obstruction that makes the ideal theory of a domain a divisibility theory rather than a decomposition theory.
Warnings
| Object | Why it is not in the list of rings with idempotents to compare | Introduced in |
|---|---|---|
| $\mathbb{O}$, the octonions | not a ring: the multiplication is not associative, so $e^2 = e$ is not defined in the ring sense | Octonion Algebra |
| the zero ring $\{0\}$ | $0 = 1$, and the single element is an idempotent, excluded by the convention $1 \neq 0$ | Rings, §2 |
| $M_2(\mathbb{R})$ as a "split" ring | it is simple, so it has no nontrivial two-sided idempotent splitting, although it has many one-sided idempotents | Matrix Algebras |
| $\mathbb{B}$ as a "split" ring | simple, like $M_2(\mathbb{C})$; its idempotents give the Peirce corners and not a product decomposition | Biquaternion Ideals and Peirce Decomposition |
The third and fourth rows record the distinction the list rests on: an idempotent gives a product decomposition only when it is central, and the idempotents of a matrix algebra are not. The split-biquaternions and the split-complex numbers are products because their idempotents are central, while the matrix ring and the biquaternions are simple and their idempotents are not.
Summary
This article has listed the rings of the corpus by their idempotents. The connected rings — the fields, the domains, the local rings, the division rings, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/p^n\mathbb{Z}$, the dual numbers and $\mathbb{F}_2[C_2]$ — have $0$ and $1$ alone; the split rings — $\mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$, the split-complex numbers, the split-biquaternions, $\mathbb{R}[x]/(x^2-1)$ and $\mathbb{Q}[C_3]$ — have four; the matrix ring and the biquaternions have infinitely many one-sided idempotents without a product decomposition, since they are simple; and the Boolean rings have one for every element. The article has recorded the direct sum decomposition $R = Re \oplus R(1-e)$ and the product decompositions it gives, the Peirce decomposition $A = \bigoplus_{i,j} e_i A e_j$ relative to a complete orthogonal family with the biquaternion corners as the worked case, and the separation of the von Neumann regular rings, in which every ideal is idempotent-generated, from the domains and local rings, in which the idempotents are scarce.
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 |
|---|---|
| $e$ | An idempotent, $e^2 = e$ |
| $e_{\pm} = \tfrac{1}{2}(1 \pm j)$ | The idempotents of the split-complex numbers and the split-biquaternions |
| $e_0 = 1$ | The identity, the trivial idempotent |
| $p = E_{11}$, $q = E_{22}$ | Orthogonal matrix-unit idempotents, $p + q = 1$ |
| $x, y$ | The nilpotent off-diagonal elements of the Peirce corners |
| $e_i A e_j$ | A Peirce corner of the algebra $A$ |
| $\omega(n)$ | The number of distinct primes dividing $n$ |
| $eM$, $(1-e)M$ | The two summands of the Peirce decomposition of a module |
Further Reading
- Tsit-Yuen Lam, A First Course in Noncommutative Rings (Springer, 2nd ed. 2001), for the idempotents, the Peirce decomposition and the corner algebras.
- Richard S. Pierce, Associative Algebras (Springer, 1982), for the Peirce decomposition treated systematically.
- Irving Kaplansky, Commutative Rings (University of Chicago Press, revised ed. 1974), for the von Neumann regular rings, the reduced rings of Krull dimension zero and the idempotents of a Noetherian ring.