List of Reduced, Local and Product Rings
Introduction
This article lists the commutative Artinian rings of the corpus in the three-way split that their structure theorem imposes: products of fields, which are reduced and semisimple; local rings, which may be non-reduced; and the mixtures, which are products of a local ring with something else. Every entry points to the article that introduces the object.
The two objects that make the split visible are the split-complex numbers $\mathbb{D} \cong \mathbb{R} \times \mathbb{R}$, which are a product of two fields, and the dual numbers $\mathbb{D}' = \mathbb{R}[\varepsilon]/(\varepsilon^2)$, which are local and non-reduced. The finite ring $\mathbb{Z}/4\mathbb{Z}$ sits with the dual numbers, and $\mathbb{Z}/12\mathbb{Z} \cong \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is the smallest mixture among the residue rings.
The article introduces nothing and proves nothing. It records examples and non-examples side by side.
The Structure Theorem
The structure theorem is that of Noetherian and Artinian Rings: a commutative Artinian ring has finitely many maximal ideals, pairwise comaximal, and is the product of its localisations at them. In particular a commutative Artinian ring is a finite product of Artinian local rings, and a reduced Artinian ring is a finite product of fields. The three families of this article are then exhaustive: a product of fields, a single local ring (the case of one factor), and a product that mixes a non-reduced local factor with other factors.
| Family | The property that defines it | Chief example | Introduced in |
|---|---|---|---|
| product of fields | reduced Artinian; every ideal is generated by an idempotent | $\mathbb{D} \cong \mathbb{R}\times\mathbb{R}$ | Split-Complex Algebra |
| local Artinian ring | a unique maximal ideal; the nonunits are the maximal ideal | $\mathbb{D}' = \mathbb{R}[\varepsilon]/(\varepsilon^2)$ | Dual-Numbers Algebra |
| mixture | a product with at least one non-reduced factor and at least two factors | $\mathbb{Z}/12\mathbb{Z}$ | Modular Arithmetic and the Ring of Residues |
The word semisimple is used in the sense of Simple and Semisimple Modules: a ring is semisimple when its regular module is semisimple, and by the Wedderburn–Artin theorem a commutative semisimple ring is exactly a finite product of fields. Every reduced Artinian ring is therefore semisimple, and a local Artinian ring that is not a field is not semisimple.
Products of Fields
| Ring | Decomposition | Idempotents | Introduced in |
|---|---|---|---|
| $\mathbb{D}$ | $\mathbb{R} \times \mathbb{R}$, via $j \mapsto (1, -1)$ | $e_{\pm} = \tfrac12(1 \pm j)$ | Split-Complex Algebra |
| $\mathbb{R} \times \mathbb{R}$ | the product itself | $(1,0)$ and $(0,1)$ | Examples of Rings and Fields |
| $\mathbb{C} \times \mathbb{C}$ | the product itself | $(1,0)$ and $(0,1)$ | Examples of Rings and Fields |
| $\mathbb{Z}/6\mathbb{Z}$ | $\mathbb{F}_2 \times \mathbb{F}_3$ | four idempotents | Modular Arithmetic and the Ring of Residues |
| $\mathbb{Z}/n\mathbb{Z}$, $n$ squarefree | $\prod_{p \mid n} \mathbb{F}_p$ | $2^{\omega(n)}$ idempotents | Modular Arithmetic and the Ring of Residues |
| $\mathbb{F}_p \times \mathbb{F}_p$ | the product itself | $(1,0)$ and $(0,1)$ | Examples of Rings and Fields |
| $\mathbb{Q}[C_2]$ | $\mathbb{Q} \times \mathbb{Q}$ | the group-ring idempotents | Examples of Rings and Fields |
| $\mathbb{Q}[C_3]$ | $\mathbb{Q} \times \mathbb{Q}(\zeta_3)$ | the group-ring idempotents | Examples of Rings and Fields |
The split-complex numbers are the geometric name for $\mathbb{R} \times \mathbb{R}$: the idempotents $\pi_+$ and $\pi_-$ are the two factors, and $j = \pi_+ - \pi_-$. This is the sense in which split-$\mathbb{C}$ sits in the reduced semisimple family: it is a product of two copies of the real field, not a field and not local.
Local Rings, Which May Be Non-Reduced
| Ring | Maximal ideal | Reduced? | Introduced in |
|---|---|---|---|
| $\mathbb{D}' = \mathbb{R}[\varepsilon]/(\varepsilon^2)$ | $(\varepsilon)$ | no: $\varepsilon^2 = 0$ | Dual-Numbers Algebra |
| $\mathbb{Z}/4\mathbb{Z}$ | $(2)$ | no: $2^2 = 0$ | Reduced Rings and the Nilradical |
| $k[x]/(x^n)$ | $(x)$ | no for $n \geq 2$ | Noetherian and Artinian Rings |
| $\mathbb{Z}/8\mathbb{Z}$ | $(2)$ | no: $2^3 = 0$ | Modular Arithmetic and the Ring of Residues |
| $\mathbb{F}_2[C_2] \cong \mathbb{F}_2[x]/(x+1)^2$ | the augmentation ideal | no | Examples of Rings and Fields |
| $\mathbb{F}_p$ | $(0)$ | yes; a field | Finite Fields |
A local Artinian ring need not be non-reduced, but every non-reduced ring of this section is local, while the local rings that are not Artinian, such as $\mathbb{Z}_{(p)}$ and $k[[x]]$, lie outside the structure theorem and are recorded in the warnings below. The dual numbers are the prototype: their maximal ideal is generated by the nilpotent $\varepsilon$, every nonunit is nilpotent, and the residue field is $\mathbb{R}$. The group ring example shows that the same shape occurs in characteristic $p$: $\mathbb{F}_2[C_2]$ is local with a nilpotent, while $\mathbb{Q}[C_2]$ is a product of two fields.
The Residue Field of Each Local Ring
Each local ring of the list is localised at its maximal ideal and carries a residue field, the quotient by that ideal. The residue field is what the local ring has in common with a field, and the nilradical is what it does not.
| Local ring | Maximal ideal | Residue field | Introduced in |
|---|---|---|---|
| $\mathbb{D}'$ | $(\varepsilon)$ | $\mathbb{R}$ | Dual-Numbers Algebra |
| $\mathbb{Z}/4\mathbb{Z}$ | $(2)$ | $\mathbb{F}_2$ | Reduced Rings and the Nilradical |
| $k[x]/(x^n)$ | $(x)$ | $k$ | Noetherian and Artinian Rings |
| $\mathbb{F}_2[C_2]$ | the augmentation ideal | $\mathbb{F}_2$ | Examples of Rings and Fields |
The residue field of a local Artinian ring is a field, and the local ring is a field exactly when its maximal ideal is $(0)$. In the split of this article a local ring is a single factor of the structure theorem, and its residue field is the field that the factor would be if it were reduced.
Mixtures
A mixture is a product with more than one factor in which at least one factor is a non-reduced local ring. The Chinese remainder theorem produces them from the residue rings, and $\mathbb{Z}/12\mathbb{Z}$ is the smallest of them.
| Ring | Decomposition | The non-reduced factor | Introduced in |
|---|---|---|---|
| $\mathbb{Z}/12\mathbb{Z}$ | $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ | $\mathbb{Z}/4\mathbb{Z}$ | Modular Arithmetic and the Ring of Residues |
| $\mathbb{Z}/n\mathbb{Z}$, $n$ not squarefree | $\prod_{p^k \parallel n} \mathbb{Z}/p^k\mathbb{Z}$ | any factor with $k \geq 2$ | Modular Arithmetic and the Ring of Residues |
The mixture is neither reduced nor local: it has nilpotents from the non-reduced factor and at least two idempotents from the product, so it is not local, and its nilradical is nonzero, so it is not reduced. It is the general case of the structure theorem, and the products of fields and the local rings are the two extremes.
Where the Three Named Objects Sit
| Object | Family | The reason | Introduced in |
|---|---|---|---|
| split-complex numbers $\mathbb{D}$ | product of fields | $\mathbb{D} \cong \mathbb{R} \times \mathbb{R}$ | Split-Complex Algebra |
| dual numbers $\mathbb{D}'$ | local, non-reduced | one maximal ideal $(\varepsilon)$, nilpotent | Dual-Numbers Algebra |
| $\mathbb{Z}/4\mathbb{Z}$ | local, non-reduced | one maximal ideal $(2)$, nilpotent | Reduced Rings and the Nilradical |
The contrast between $\mathbb{D}$ and $\mathbb{D}'$ is the whole content of the split: the quotient $\mathbb{R}[x]/(f)$ of the polynomial ring by a quadratic is a product of two copies of $\mathbb{R}$ when $f$ has two distinct real roots, the field $\mathbb{C}$ when $f$ is irreducible, and a local ring with a nilpotent when $f$ has a double root. Thus $j^2 - 1$ gives the product and $\varepsilon^2$ gives the local ring, as recorded in Examples of Rings and Fields.
Warnings
| Object | Why it is not in this split | Introduced in |
|---|---|---|
| $M_n(\mathbb{R})$, $n \geq 2$ | semisimple and simple as a ring, but not commutative and not a product of fields | Matrix Algebras |
| $\mathbb{H}$ | a division ring, finite-dimensional over $\mathbb{R}$, but not commutative | Quaternion Algebra |
| $\mathbb{B}$, $\mathbb{H}_{\mathbb{D}}$ | not commutative, and with zero divisors; not in the commutative Artinian split | Biquaternion Algebra, Split-Biquaternion Algebra |
| $\mathbb{O}$ | not a ring | Octonion Algebra |
| $\mathbb{Z}$, $\mathbb{Z}[x]$, $k[x]$ | commutative and reduced, but not Artinian, so the structure theorem does not apply | The Integers, Polynomial Rings and Rational Functions |
| $\mathbb{Z}_{(p)}$, $k[[x]]$ | local and reduced, but not Artinian: a discrete valuation ring has dimension $1$ | Localization and the Fraction Field, Examples of Rings and Fields |
Summary
This article has listed the commutative Artinian rings of the corpus in the three families of the structure theorem. The products of fields contain split-$\mathbb{C}$, the squarefree residue rings and the rational group rings $\mathbb{Q}[C_2]$ and $\mathbb{Q}[C_3]$; the local rings contain the dual numbers, $\mathbb{Z}/4\mathbb{Z}$ and the truncated polynomial rings, and may be reduced or not; the mixtures contain $\mathbb{Z}/12\mathbb{Z}$ and the residue rings with a repeated prime factor. The three objects the scope names sit as follows: split-$\mathbb{C}$ with the products of fields, the dual numbers and $\mathbb{Z}/4\mathbb{Z}$ with the local non-reduced rings.
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 |
|---|---|
| $\mathbb{D}$ | The split-complex numbers, $\mathbb{R}[j]/(j^2-1) \cong \mathbb{R}\times\mathbb{R}$ |
| $\mathbb{D}'$ | The dual numbers, $\mathbb{R}[\varepsilon]/(\varepsilon^2)$ |
| $e_{\pm}$ | The idempotents $\tfrac12(1 \pm j)$ of $\mathbb{D}$ |
| $\mathrm{M}$ | The maximal ideal of a local ring |
| $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{F}_p$ | Residue ring, prime field |
| $k[x]/(x^n)$, $\mathbb{F}_2[C_2]$ | Truncated polynomial ring, a local group ring |
| $\mathbb{Z}_{(p)}$, $k[[x]]$ | Localisations and power series rings, discrete valuation rings |
| $M_n(\mathbb{R})$, $\mathbb{H}$, $\mathbb{B}$, $\mathbb{H}_{\mathbb{D}}$, $\mathbb{O}$ | Non-commutative or non-associative objects, named in the warnings |
| $\operatorname{nil}(R)$ | The nilradical of $R$ |
Further Reading
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (Addison–Wesley, 1969), for the structure theorem for Artinian rings and the primacy of the local case.
- Irving Kaplansky, Commutative Rings (University of Chicago Press, revised ed. 1974), for products of fields, local rings and the idempotent decompositions of a product.
- Tsit-Yuen Lam, Exercises in Classical Ring Theory (Springer, 2nd ed. 2003), for the dual numbers, $\mathbb{Z}/4\mathbb{Z}$ and the group-ring counterexamples.