List of Artinian and Noetherian Rings
Introduction
This article lists the rings the corpus meets that satisfy the ascending chain condition or the descending chain condition on ideals. A ring is Noetherian when every ascending chain of ideals stabilises, equivalently when every ideal is finitely generated; it is Artinian when every descending chain stabilises, equivalently when it has finite length. Artinian is the stronger condition, by the Akizuki–Hopkins–Levitzki theorem, and the semisimple Artinian rings are described by Wedderburn–Artin.
Every entry points to the article that introduces the ring or the condition, together with the chain condition it satisfies. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being a ring that has one chain condition but fails the other, with the failure named and the article that records it.
The Two Chain Conditions
The ascending chain condition (ACC) on ideals says that every ascending chain $I_0 \subseteq I_1 \subseteq \cdots$ stabilises, equivalently that every ideal is finitely generated; the descending chain condition (DCC) is the mirror statement. The conditions are inherited by quotients and localisations, and the Hilbert basis theorem gives the polynomial rings. A ring of finite length has a composition series, and the length is additive in short exact sequences.
| Object | The property it has | Introduced in |
|---|---|---|
| Ascending chain condition (ACC) | every ascending chain of ideals stabilises; equivalently every ideal is finitely generated | Noetherian and Artinian Rings |
| Descending chain condition (DCC) | every descending chain of ideals stabilises | Noetherian and Artinian Rings |
| Noetherian ring | a ring with the ACC | Noetherian and Artinian Rings |
| Artinian ring | a ring with the DCC | Noetherian and Artinian Rings |
| Akizuki–Hopkins–Levitzki theorem | every Artinian ring is Noetherian | Noetherian and Artinian Rings |
| Closure under quotients | a quotient of a Noetherian or Artinian ring is again Noetherian or Artinian | Noetherian and Artinian Rings |
| Closure under localisation | a localisation of a Noetherian ring is Noetherian | Localization and the Fraction Field |
| Composition series and length | a ring of finite length has a composition series with a well-defined length $\ell(R)$ | Noetherian and Artinian Rings |
| Krull dimension $0$ | every prime ideal is maximal; the Artinian rings are exactly those that are Noetherian of dimension $0$ | Integral Extensions and Krull Dimension |
| Non-example: a ring with neither chain condition | fails both conditions: an infinite-dimensional algebra with a non-finitely generated ideal | Noetherian and Artinian Rings |
Noetherian Rings
The Noetherian condition is the stable one: it is preserved by quotients, localisations, finite products and polynomial extension, and it holds for the rings the corpus computes with. The Hilbert basis theorem states that $R[x]$ is Noetherian when $R$ is, so the polynomial ring in finitely many variables over a field is Noetherian; the ring is nevertheless not Artinian, since the chain $(x) \supsetneq (x^2) \supsetneq \cdots$ does not stabilise.
| Ring | The property it has | Introduced in |
|---|---|---|
| Field $F$ | Noetherian and Artinian: its only ideals are $0$ and $F$ | Fields |
| $\mathbb{Z}$ | Noetherian, not Artinian; a principal ideal domain | Noetherian and Artinian Rings |
| $\mathbb{Z}/n\mathbb{Z}$ | Noetherian and Artinian: a finite ring | Noetherian and Artinian Rings |
| $R[x_1,\dots,x_n]$ | Noetherian, by the Hilbert basis theorem applied $n$ times | Noetherian and Artinian Rings |
| Principal ideal domain | Noetherian: every ideal is generated by one element | Principal Ideal Domains |
| Dedekind domain | Noetherian of Krull dimension $1$; every nonzero ideal a product of primes | Dedekind Domains and Ideal Class Groups |
| $S^{-1}R$ | Noetherian when $R$ is; the localisation at a multiplicative set | Localization and the Fraction Field |
| $R/I$ | Noetherian when $R$ is; the quotient by an ideal | Noetherian and Artinian Rings |
| Finitely generated module over a Noetherian ring | Noetherian as a module: submodules are finitely generated | Modules |
| Non-example: $\mathbb{Z}$ | fails the DCC: the chain $(2) \supsetneq (4) \supsetneq (8) \supsetneq \cdots$ does not stabilise | Noetherian and Artinian Rings |
| Non-example: $k[x]$ | fails the DCC: the chain $(x) \supsetneq (x^2) \supsetneq \cdots$ does not stabilise | Noetherian and Artinian Rings |
Artinian Rings
An Artinian ring is a finite product of Artinian local rings, and every Artinian ring is Noetherian by Akizuki–Hopkins–Levitzki and of Krull dimension $0$. The standard Artinian rings are the fields, the finite rings and the local rings $k[x]/(x^n)$; the dual numbers $\mathbb{D}'_F = F[x]/(x^2)$ are the smallest non-reduced example.
| Ring | The property it has | Introduced in |
|---|---|---|
| Field $F$ | Artinian of length $1$, of Krull dimension $0$ | Fields |
| $\mathbb{Z}/n\mathbb{Z}$ | Artinian and Noetherian; a product of local rings $\mathbb{Z}/p_i^{e_i}\mathbb{Z}$ | Noetherian and Artinian Rings |
| $k[x]/(x^n)$ | local Artinian of length $n$, with nilpotent maximal ideal | Noetherian and Artinian Rings |
| Dual numbers $\mathbb{D}'_F = F[x]/(x^2)$ | local Artinian, non-reduced, of length $2$ | Noetherian and Artinian Rings |
| Artinian local ring | a local ring of finite length; the factors in the product decomposition | Noetherian and Artinian Rings |
| Finite product of Artinian local rings | the general Artinian ring, by the structure theorem | Noetherian and Artinian Rings |
| Reduced Artinian ring | a finite product of fields; the reduced rings of Krull dimension $0$ | Reduced Rings and the Nilradical |
| Non-example: $\mathbb{Z}$ | fails the DCC, so fails to be Artinian | Noetherian and Artinian Rings |
| Non-example: $k[x]$ | fails the DCC, and has Krull dimension $1$ | Noetherian and Artinian Rings |
| Non-example: $k[x_1,\dots,x_n]$ for $n \geq 1$ | fails the DCC, and has Krull dimension $n$ | Integral Extensions and Krull Dimension |
Semisimple and Simple Artinian Rings
A ring is semisimple when its regular module is a direct sum of simples, equivalently when every module over it is semisimple; the Wedderburn–Artin theorem writes such a ring as a product $\prod_i M_{n_i}(D_i)$ of matrix rings over division rings. Semisimple rings are Artinian and Noetherian of Krull dimension $0$, and a simple Artinian ring is a single matrix ring over a division ring. This is where the non-commutative examples of the corpus sit.
| Ring | The property it has | Introduced in |
|---|---|---|
| Wedderburn–Artin decomposition | a semisimple ring is $\prod_i M_{n_i}(D_i)$, the factors determined up to permutation | Simple and Semisimple Modules |
| $\mathbb{H}$ | a division ring; simple Artinian and Noetherian, of Krull dimension $0$, semisimple | Division Algebras |
| $M_2(\mathbb{R})$ | simple Artinian and Noetherian; the matrix ring $M_2(\mathbb{R})$, of length $2$ over $\mathbb{R}$ | Examples of Algebras |
| Biquaternions $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ | isomorphic to $M_2(\mathbb{C})$ over $\mathbb{C}$; simple Artinian and Noetherian | Biquaternion Algebra |
| Split-complex numbers $\mathbb{D} = \mathbb{R}[t]/(t^2-1)$ | isomorphic to $\mathbb{R}\times\mathbb{R}$; reduced Artinian of dimension $0$, not a field | Split-Complex Algebra |
| Dual numbers $\mathbb{D}'$ | local Artinian, non-reduced; the smallest non-semisimple Artinian ring | Dual Numbers Algebra |
| Non-example: $\mathbb{B}$ treated as a field | fails: it has zero divisors, so it is not a division ring, though it is Artinian | Biquaternion Algebra |
| Non-example: $M_2(\mathbb{R})$ treated as a division ring | fails: a nonzero nilpotent matrix has no inverse | Examples of Algebras |
The Two Conditions Compared
The two conditions are not symmetric: DCC implies ACC by Akizuki–Hopkins–Levitzki, but not conversely, and the examples separate the cells. The table records where each standard ring sits.
| Ring | ACC (Noetherian) | DCC (Artinian) | The invariant that shows it | Introduced in |
|---|---|---|---|---|
| Field $F$ | yes | yes | length $1$, dimension $0$ | Fields |
| $\mathbb{Z}/n\mathbb{Z}$ | yes | yes | finite, length $\Omega(n)$ | Noetherian and Artinian Rings |
| $M_n(D)$ over a division ring | yes | yes | simple Artinian, length $n$ | Simple and Semisimple Modules |
| $\mathbb{Z}$ | yes | no | the chain $(2) \supsetneq (4) \supsetneq \cdots$, dimension $1$ | Noetherian and Artinian Rings |
| $k[x]$ | yes | no | the chain $(x) \supsetneq (x^2) \supsetneq \cdots$, dimension $1$ | Noetherian and Artinian Rings |
| $k[x_1,\dots,x_n]$, $n \geq 1$ | yes | no | dimension $n$ by Krull's theorem | Integral Extensions and Krull Dimension |
| Dedekind domain | yes | no | dimension $1$; the nonzero primes are maximal | Dedekind Domains and Ideal Class Groups |
| A Noetherian ring of dimension $0$ | yes | yes | Artinian by the structure theorem | Integral Extensions and Krull Dimension |
| A ring of finite length | yes | yes | the length is additive in short exact sequences | Noetherian and Artinian Rings |
| $k[x]/(x^n)$ | yes | yes | local Artinian of length $n$, non-reduced for $n \geq 2$ | Noetherian and Artinian Rings |
| $\mathbb{D} \cong \mathbb{R}\times\mathbb{R}$ | yes | yes | reduced Artinian of dimension $0$ | Split-Complex Algebra |
The one-way implication is the content of Akizuki–Hopkins–Levitzki, and the table shows its sharpness: every Artinian row has both conditions, while $\mathbb{Z}$, $k[x]$ and the Dedekind domains have the ACC alone. The Artinian rows are exactly the Noetherian rows of Krull dimension $0$, so the two conditions differ precisely by the dimension.
Summary
The list gathers the Artinian and Noetherian rings. The Noetherian examples are the fields, $\mathbb{Z}$, the finite rings $\mathbb{Z}/n\mathbb{Z}$, the polynomial rings $R[x_1,\dots,x_n]$, the principal ideal domains, the Dedekind domains, the localisations and the quotients, together with the finitely generated modules over a Noetherian ring; the Artinian examples are the fields, the finite rings, the local rings $k[x]/(x^n)$, the dual numbers and the finite products of Artinian local rings. The passage between the two is Akizuki–Hopkins–Levitzki, the length and Krull dimension $0$ are the measuring invariants, and the semisimple rings are the products $\prod_i M_{n_i}(D_i)$ of matrix rings over division rings. The non-commutative examples sit here: $\mathbb{H}$ is a division ring, $M_2(\mathbb{R})$ and the biquaternions $\mathbb{B}$ are simple Artinian matrix rings, the split-complex numbers are the reduced Artinian ring $\mathbb{R}\times\mathbb{R}$ and the dual numbers are local Artinian and non-reduced. The non-examples — $\mathbb{Z}$ and $k[x]$ and $k[x_1,\dots,x_n]$ as Noetherian but not Artinian, $\mathbb{B}$ mistaken for a field and $M_2(\mathbb{R})$ mistaken for a division ring — name the property that fails.
Summary of Notation
The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.
| Symbol | Meaning |
|---|---|
| ACC, DCC | ascending and descending chain condition on ideals |
| $\ell(R)$, $\dim R$ | length and Krull dimension of a ring |
| $R[x_1,\dots,x_n]$ | polynomial ring; Noetherian by the Hilbert basis theorem |
| $S^{-1}R$ | localisation of a ring |
| $M_n(D)$ | matrix ring over a division ring |
| $\mathbb{H}$, $\mathbb{B}$ | quaternions and biquaternions |
| $\mathbb{D}$, $\mathbb{D}'$ | split-complex numbers and dual numbers |
| $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$ | integers and their quotients |
| $\mathbb{R}$, $\mathbb{C}$ | the reals and the complex numbers |
| $\prod_i M_{n_i}(D_i)$ | Wedderburn–Artin decomposition |
Further Reading
- Oscar Zariski and Pierre Samuel, Commutative Algebra, Vol. I (Van Nostrand, 1958), for the chain conditions, the Hilbert basis theorem and the structure of Artinian rings.
- Michael Atiyah and Ian Macdonald, Introduction to Commutative Algebra (Addison–Wesley, 1969), for the chain conditions, Krull dimension and the Akizuki–Hopkins–Levitzki theorem.
- Frank Anderson and Kent Fuller, Rings and Categories of Modules (Springer, 2nd ed. 1992), for the Wedderburn–Artin theorem and the simple and semisimple Artinian rings.
- Irving Kaplansky, Rings of Operators (Benjamin, 1968), for the finite-dimensional and von Neumann algebraic contrasts with the Artinian structure theory.