Semiperfect Rings with Simple Quotient and Commutative Semiperfect Rings

Q: Is k in (23.10) canonically attached to R?

Up to isomorphism, yes; canonically as a subring, no. k ≅ e_1Re_1 depends on the choice of e_1, and different choices are conjugate by a unit. What is intrinsic is the isomorphism class, pinned down by Krull–Schmidt applied to R_R.

Q: Why can't (23.10) be extended to arbitrary semiperfect rings?

Because the product decomposition of R is carried by central idempotents, and semiperfectness only guarantees that idempotents lift. T_2(k) has R ≅ k × k yet is indecomposable, since its centre is just the scalars. Recovering a product decomposition requires the block theory of §25.

Q: Where is commutativity actually used in (23.11)?

In exactly one place: it makes the local idempotents e_i central, so that the module decomposition R = e_iR is a ring decomposition. Without it, e_iR is only a right ideal and there is no product.

Q: How is Akizuki–Cohen related to the Hopkins–Levitzki theorem?

Hopkins–Levitzki supplies the implication artinian noetherian in general, and the commutative case was obtained earlier by Akizuki. The hard direction of (23.12) is the converse, noetherian plus dimension zero artinian, proved by the filtration by products of maximal ideals.

Executive Summary

Wedderburn–Artin says a simple artinian ring is $M_{n} (D)$ for a division ring $D$ . This page proves the exact analogue one level up: a semiperfect ring whose radical quotient is simple is $M_{n} (k)$ for a local ring $k$ , with $n$ and $k$ determined. Passing from a division ring to a local ring is precisely the price of allowing a radical.

The commutative case is even cleaner: a commutative ring is semiperfect if and only if it is a finite direct product of local rings. Combined with the Hopkins–Levitzki theorem this yields the classical Akizuki–Cohen characterisation — a commutative ring is artinian iff it is noetherian of Krull dimension zero.

$M_{n} (k)$ Simple quotient case

$\prod_{i} k_{i}$ Commutative case

(23.10)Wedderburn–Artin analogue

(23.12)Akizuki–Cohen

Overview

For a general semiperfect ring $R$ , the quotient $\bar{R} = R / rad R$ is a finite product of simple artinian rings. Two special cases admit a complete structure theory.

\bar{R}

simple

R ≅ M_{n} (k)

k

local

R

indecomposable as a ring

The first case, $(23.10)$ , applies when $\bar{R}$ has a single Wedderburn factor. Equivalently, all the principal indecomposables $R e_{1}, \dots, R e_{n}$ of $(23.6)$ are isomorphic to one another, so $R$ is a matrix ring over the endomorphism ring of any one of them.

The second case is commutativity. Here every idempotent is central, so the decomposition of $1$ into orthogonal local idempotents is at once a decomposition of the ring into a product of local rings — that is $(23.11)$ .

$(23.10)$ should be read as the semiperfect extension of Wedderburn–Artin. Both statements say one simple factor means one matrix ring; the difference is that the coefficient ring is a division ring in the semisimple case and merely local in the semiperfect one.

Learning Objectives

Prove that $M_{n} (k)$ with $k$ local is semiperfect with simple radical quotient and is indecomposable as a ring.
Prove the converse, constructing $k$ as the endomorphism ring of a principal indecomposable.
State the uniqueness of $n$ and of $k$ and justify it by Krull–Schmidt.
Prove that commutative semiperfect rings are exactly finite products of local rings.
Derive the Akizuki–Cohen equivalences for commutative artinian rings.
Give an indecomposable semiperfect ring whose radical quotient is decomposable.

Definitions

$\bar{R}$ simple: $\bar{R}$ has no proper nonzero two-sided ideals. Together with semiperfectness this forces $\bar{R}$ to be simple artinian, i.e. $M_{n} (D)$ for a division ring $D$ .
Local ring $k$: $k / rad k$ is a division ring; equivalently the non-units of $k$ form an ideal.
Ring decomposition: $R ≅ A \times B$ with $A, B$ nonzero; equivalent to the existence of a central idempotent other than $0$ and $1$ .
Krull dimension zero: For a commutative ring, every prime ideal is maximal. Equivalently, the ring has no strict inclusions between primes.
Artinian local ring: A local ring satisfying the descending chain condition; its maximal ideal is nilpotent and it has finite length over itself.

In $(23.12)$ , artinian and noetherian are unambiguous because the ring is commutative; elsewhere on this page the semiperfect hypothesis is side-neutral by definition.

Key Results

Theorem(23.10)Semiperfect rings with simple radical quotient

For a ring $R$ the following are equivalent:

$R$ is semiperfect and $R / rad R$ is a simple ring;
$R ≅ M_{n} (k)$ for some $n \geq 1$ and some local ring $k$ .

When these hold, $n$ is uniquely determined and $k$ is unique up to isomorphism. Moreover $R$ is indecomposable as a ring.

Proof

**(2) $\Rightarrow$ (1).** Let $k$ be local with $D = k / rad k$ . Then $R = M_{n} (k)$ is semiperfect by $(23.2)$ , and $R / rad R ≅ M_{n} (D)$ is simple artinian, hence simple.

**(1) $\Rightarrow$ (2).** By $(23.6)$ write $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ mutually orthogonal local idempotents, so that $R_{R} = e_{1} R \oplus \dots \oplus e_{n} R$ and $\bar{R} = {\bar{e}}_{1} \bar{R} \oplus \dots \oplus {\bar{e}}_{n} \bar{R}$ with each ${\bar{e}}_{i} \bar{R}$ a minimal right ideal. Since $\bar{R}$ is simple artinian it has a unique simple right module up to isomorphism, so all the ${\bar{e}}_{i} \bar{R}$ are isomorphic. Isomorphism of idempotent-generated right ideals is detected modulo the radical — $e_{i} R ≅ e_{j} R$ if and only if ${\bar{e}}_{i} \bar{R} ≅ {\bar{e}}_{j} \bar{R}$ — so all the $e_{i} R$ are isomorphic. Writing $M = e_{1} R$ we get $R_{R} ≅ M^{n}$ and hence

R ≅ End (R_{R}) ≅ End (M^{n}) ≅ M_{n} (End (M)) = M_{n} (k), k : = End (M) ≅ e_{1} R e_{1},

(23.10a)

and $k$ is local because $e_{1}$ is a local idempotent.

Uniqueness. Suppose $R ≅ M_{n} (k)$ with $k$ local. Then $R_{R}$ is a direct sum of $n$ copies of the column module $M$ , each with local endomorphism ring $k$ . By the Krull–Schmidt–Azumaya theorem such a decomposition of $R_{R}$ is unique up to isomorphism and permutation, so $n$ is the number of indecomposable summands of $R_{R}$ and $k ≅ End (M)$ is determined up to isomorphism.

Indecomposability. The centre of $M_{n} (k)$ consists of the scalar matrices $c I$ with $c \in Z (k)$ . A central idempotent of $R$ therefore has the form $c I$ with $c = c^{2} \in k$ ; a local ring has no idempotents besides $0$ and $1$ , so $c \in {0, 1}$ and $R$ has only the trivial central idempotents. Hence $R$ is indecomposable as a ring.

Remark—Why there is no general version

If $R$ is semiperfect but $\bar{R}$ has several Wedderburn factors, $\bar{R} ≅ \prod_{j} M_{n_{j}} (D_{j})$ , one would like to lift the corresponding centrally primitive idempotents of $\bar{R}$ to central idempotents of $R$ and split $R$ accordingly. This fails: $(23.1)$ guarantees lifting of idempotents, not of central idempotents. The upper triangular ring $T_{2} (k)$ is indecomposable with $\bar{R} ≅ k \times k$ .

Theorem(23.11)Commutative semiperfect rings

A commutative ring $R$ is semiperfect if and only if $R$ is a finite direct product of (commutative) local rings.

Proof

( $\Leftarrow$ ) Every local ring is semiperfect, and a finite direct product of semiperfect rings is semiperfect by $(23.4)$ .

( $\Rightarrow$ ) Let $R$ be commutative and semiperfect, and take $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ orthogonal local idempotents, as in $(23.6)$ . In a commutative ring every element is central, so each $e_{i}$ is a central idempotent and the decomposition of the module $_{R} R$ is a decomposition of the ring:

R ≅ e_{1} R \times e_{2} R \times \dots \times e_{n} R .

(23.11a)

Each factor $e_{i} R = e_{i} R e_{i}$ is local because $e_{i}$ is a local idempotent.

Hence $R$ is a finite product of local rings. Note the number of factors and the factors themselves are then determined: the $e_{i}$ are the centrally primitive idempotents of $R$ , which in the commutative case are canonical.

Corollary(23.12)Akizuki–Cohen

For a commutative ring $R$ the following are equivalent:

$R$ is artinian;
$R$ is a finite direct product of artinian local rings;
$R$ is noetherian and has Krull dimension $0$ , i.e. every prime ideal of $R$ is maximal.

Proof

**(1) $\Leftrightarrow$ (2).** An artinian commutative ring is semiperfect, so $(23.11)$ presents it as a finite product of local rings, and each factor — a quotient of $R$ — is artinian. Conversely a finite product of artinian rings is artinian.

**(1) $\Rightarrow$ (3).** Noetherianness is the Hopkins–Levitzki theorem. For dimension zero, let $𝔭$ be prime; then $R / 𝔭$ is an artinian integral domain. For $0 \neq a \in R / 𝔭$ the chain $(a) \supseteq (a^{2}) \supseteq \dots$ stabilises, so $a^{m} = a^{m + 1} b$ for some $b$ , and cancelling $a^{m}$ in the domain gives $a b = 1$ . So $R / 𝔭$ is a field and $𝔭$ is maximal.

**(3) $\Rightarrow$ (1).** Assume $R \neq 0$ is noetherian of dimension $0$ . In a noetherian ring every ideal contains a finite product of prime ideals; applied to the zero ideal this gives primes — hence, by dimension $0$ , maximal ideals — with $𝔪_{1} 𝔪_{2} \dots 𝔪_{r} = 0$ . Consider the filtration

0 = 𝔪_{1} \dots 𝔪_{r} \subseteq 𝔪_{1} \dots 𝔪_{r - 1} \subseteq \dots \subseteq 𝔪_{1} \subseteq R .

(23.12a)

Each successive quotient $𝔪_{1} \dots 𝔪_{i} / 𝔪_{1} \dots 𝔪_{i + 1}$ is annihilated by $𝔪_{i + 1}$ , hence is a vector space over the field $R / 𝔪_{i + 1}$ , and it is finitely generated because $R$ is noetherian. A finite-dimensional vector space has a composition series, so $R$ itself has finite length as an $R$ -module and is therefore artinian.

Remark—A noncommutative analogue

The equivalence (1) $\Leftrightarrow$ (3) survives without commutativity in the following form: a ring $R$ is right artinian if and only if $R$ is right noetherian and $R / 𝔭$ is simple artinian for every prime ideal $𝔭 \subseteq R$ . The proof follows the commutative one, with simple artinian replacing field.

Proof Techniques and Method

How these proofs work, and which move is reusable.

Move 1

Count isomorphism classes downstairs

A simple quotient means one simple module, hence all principal indecomposables are isomorphic upstairs. Recognising a ring as $M_{n} (k)$ is always a statement about how many isomorphism classes of summands $R_{R}$ has.

Move 2

Detect isomorphism modulo the radical

For idempotents $e, f$ : $e R ≅ f R$ iff $\bar{e} \bar{R} ≅ \bar{f} \bar{R}$ . This is what transports the semisimple computation back to $R$ , and it is used in almost every semiperfect structure proof.

Move 3

Use commutativity to centralise

Once idempotents are central, a module decomposition is a ring decomposition. Everything special about the commutative case comes from this single observation.

The Akizuki–Cohen proof adds a fourth, purely commutative technique: build a filtration by products of maximal ideals and read off finite length one layer at a time. The same argument shows that a noetherian ring in which some product of maximal ideals vanishes is artinian, which is the form usually needed in practice.

Worked Example

A matrix ring over a local ring

Take $k = ℤ_{p}$ and $R = M_{2} (ℤ_{p})$ . Then $rad R = M_{2} (p ℤ_{p})$ and $\bar{R} ≅ M_{2} (𝔽_{p})$ , which is simple. By $(23.10)$ the pair $(n, k) = (2, ℤ_{p})$ is determined by $R$ : any isomorphism $R ≅ M_{m} (k')$ with $k'$ local forces $m = 2$ and $k' ≅ ℤ_{p}$ .

Indecomposability is visible: $Z (R) = ℤ_{p} \cdot I$ , and $ℤ_{p}$ has no idempotents besides $0$ and $1$ , so $R$ is not a product of two nonzero rings — even though $R$ is far from being a domain.

The commutative case: $ℤ / 12$

$R = ℤ / 12$ is commutative artinian, hence semiperfect. Its maximal ideals are $(2)$ and $(3)$ , so $rad R = (6)$ and $\bar{R} ≅ 𝔽_{2} \times 𝔽_{3}$ . The idempotents of $\bar{R}$ do lift here, to $e_{1} = 9$ and $e_{2} = 4$ :

9^{2} = 81 = 12 \cdot 6 + 9 \equiv 9, 4^{2} = 16 \equiv 4, 9 \cdot 4 = 36 \equiv 0, 9 + 4 = 13 \equiv 1 (\mod 12) .

(E.1)

An orthogonal pair of idempotents summing to $1$ — exactly the decomposition promised by $(23.6)$ and $(23.11)$ .

Hence $ℤ / 12 ≅ 9 ℤ / 12 \times 4 ℤ / 12 ≅ ℤ / 4 \times ℤ / 3$ , a product of two artinian local rings, illustrating $(23.11)$ and clause (2) of $(23.12)$ .

Indecomposable with a decomposable quotient

Let $k$ be a field and $R = T_{2} (k)$ , the upper triangular $2 \times 2$ matrices. Then $rad R$ is the strictly upper triangular part and $\bar{R} ≅ k \times k$ , which is not simple, so $(23.10)$ does not apply. And indeed:

Z (T_{2} (k)) = {(\begin{matrix} a & 0 \\ 0 & a \end{matrix}) : a \in k} ≅ k ⟹ T_{2} (k) has only the central idempotents 0, 1 .

(E.2)

The central idempotent $(1, 0)$ of $\bar{R}$ lifts to $E_{11}$ , which is idempotent but not central.

So $T_{2} (k)$ is an indecomposable semiperfect ring whose radical quotient decomposes. This is the precise obstruction referred to in the Remark above, and it is why block theory is a separate development rather than a corollary.

Comparison and Classification

Structure of a semiperfect ring by the shape of $\bar{R}$
$\bar{R} = R / rad R$	Extra hypothesis	Structure of $R$	Reference
$D$ , a division ring	none	$R$ is local	definition
$M_{n} (D)$ , simple	none	$R ≅ M_{n} (k)$ , $k$ local, $R$ indecomposable	(23.10)
$\prod_{j} D_{j}$ , product of fields	$R$ commutative	$R ≅ \prod_{j} k_{j}$ , each $k_{j}$ local	(23.11)
$\prod_{j} M_{n_{j}} (D_{j})$	none	no product decomposition in general	Remark, $T_{2} (k)$
$\prod_{j} M_{n_{j}} (D_{j})$	central idempotents lift	$R ≅ \prod_{j} R_{j}$ with each ${\bar{R}}_{j}$ simple	block theory, §25

Commutative rings against the three Akizuki–Cohen conditions
	Noetherian	Dimension $0$	Artinian	Semiperfect
$ℤ / 12$	yes	yes	yes	yes
a field $k$	yes	yes	yes	yes
$k [x] / (x^{3})$	yes	yes	yes	yes
$ℤ$	yes	no	no	no
$k [[x]]$	yes	no	no	yes
$ℤ$ localised away from $2, 3$	yes	no	no	no
$k [x_{1}, x_{2}, \dots] / (x_{1}, x_{2}, \dots)^{2}$	no	yes	no	yes

Commutative rings against the three Akizuki–Cohen conditions

The two yes entries in the last column with no in the artinian column show how much wider semiperfect is than artinian even among commutative rings: $k [[x]]$ is local, and the last ring is local with a square-zero maximal ideal.

Relationship Map

Semiperfect ring $R$
- $\bar{R}$ simple
  - $R ≅ M_{n} (k)$ , $k$ local
  - $n$ and $k$ unique
  - $R$ indecomposable as a ring
- $R$ commutative
  - $R ≅ k_{1} \times \dots \times k_{n}$ , each $k_{i}$ local
  - the $e_{i}$ are the centrally primitive idempotents
  - artinian iff every $k_{i}$ is artinian
- general case
  - $\bar{R}$ is a finite product of simple artinian rings
  - central idempotents may fail to lift
  - block decomposition treated separately

Commutative artinianCommutative semiperfectCommutative semilocal

The first arrow is strict — $k [[x]]$ is semiperfect and not artinian — and so is the second, since a semilocal domain with two maximal ideals is not semiperfect. See Semiperfect Rings: Definition and First Examples for that witness.

Your semiperfect ring $R$ : what structure can you claim?

$\bar{R}$ is a division ring

R

is local. This is the

n = 1

case of

(23.10)

$\bar{R}$ is simple

R ≅ M_{n} (k)

with

k

local;

n

and

k

are invariants, and

R

is indecomposable.

$R$ is commutative

R

is a finite product of local rings, with the factors canonically determined.

NeitherOnly the general theory applies: a decomposition of

1

into orthogonal local idempotents, and a finite list of simple modules. Any product decomposition must be found by exhibiting central idempotents.

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Commutative algebra

Zero-dimensional rings

$(23.12)$ is the standard reason that an artinian ring is a finite product of local rings, which underlies primary decomposition, the structure of fibres of finite morphisms, and the definition of intersection multiplicity.

Number theory

Reduction of orders

For a maximal order in a division algebra over a $p$ -adic field, the completion is $M_{n} (k)$ with $k$ local; $(23.10)$ pins down $n$ and $k$ from the residue data and is the local input to classification by Hasse invariants.

Coding theory

Codes over finite chain rings

A finite commutative ring is artinian, hence by $(23.12)$ a product of local rings; code constructions therefore reduce, coordinate by coordinate, to the local case with its residue field.

Computer algebra

Splitting a finite ring

Systems compute the idempotent decomposition of a finite commutative ring to reduce arithmetic to local factors — the same Chinese-Remainder step that $(23.11)$ formalises for arbitrary commutative semiperfect rings.

Honest summary: $(23.10)$ and $(23.11)$ are recognition theorems. Their value is that they turn a hypothesis you can check into an isomorphism you can compute with.

Failure Modes and Common Mistakes

$(23.11)$ needs finite: an infinite product of fields is commutative, has all idempotents you could want, and is not semiperfect because it is not semilocal.
The local ring $k$ in $(23.10)$ is not a subring of $R$ in any canonical way — it is $e_{1} R e_{1}$ for a choice of $e_{1}$ , determined only up to isomorphism.
Do not read $(23.10)$ as saying every indecomposable semiperfect ring is a matrix ring over a local ring. $T_{2} (k)$ is indecomposable and is not of that form.
In $(23.12)$ , *Krull dimension $0$ * means every prime is maximal, not that the ring has a unique prime. A product of $r$ fields has $r$ primes and dimension $0$ .

Historical Notes and Lessons Learned

1908WedderburnStructure theorem for finite-dimensional algebras: a simple algebra is a matrix algebra over a division algebra. This is the shape $(23.10)$ reproduces one level up.
1927ArtinExtension of the structure theory to rings with the descending chain condition, making simple artinian the right hypothesis rather than finite-dimensional.
1935AkizukiCommutative rings that are noetherian of dimension zero are shown to be artinian, the implication that is hardest in $(23.12)$ .
1939Hopkins and LevitzkiA one-sided artinian ring with identity is one-sided noetherian, supplying the reverse implication in the noncommutative setting.
1940sCohenCohen's work on the ideal theory of local rings puts the zero-dimensional characterisation into its standard commutative-algebra form.
1960BassSemiperfect rings are introduced, and results such as $(23.10)$ are recognised as the natural extension of Wedderburn–Artin beyond the semisimple case.

The lesson is that each generalisation replaced a hypothesis about the ring by a hypothesis about its radical quotient plus a lifting property. Once that is the frame, $(23.10)$ is forced: a single simple factor downstairs can only come from a matrix ring over something with a unique maximal one-sided ideal.

Quick Reference

$(23.10)$

R

semiperfect with

\bar{R}

simple

iff R ≅ M_{n} (k)

k

local

Invariants

n

= number of indecomposable summands of

R_{R}

;

k ≅ e_{1} R e_{1}

Consequencesuch an

R

is indecomposable as a ring

$(23.11)$

R

commutative semiperfect

iff R ≅ k_{1} \times \dots \times k_{n}

, each local

$(23.12)$ commutative: artinian

iff

product of artinian local rings

iff

noetherian and

\dim = 0

Obstructioncentral idempotents of

\bar{R}

need not lift; witness

T_{2} (k)

Noncommutative analogueright artinian

iff

right noetherian and

R / 𝔭

simple artinian for all primes

𝔭

Special case

n = 1

(23.10)

recovers *

R

local*

Results of this page
Reference	Statement	Hypotheses
(23.10)	$R ≅ M_{n} (k)$ with $k$ local; $n$ , $k$ unique; $R$ indecomposable	$R$ semiperfect and $R / rad R$ simple
(23.11)	$R$ is a finite product of local rings	$R$ commutative and semiperfect
(23.12)	Artinian $iff$ product of artinian local rings $iff$ noetherian of dimension $0$	$R$ commutative
Remark	Right artinian $iff$ right noetherian with all $R / 𝔭$ simple artinian	$𝔭$ ranges over prime ideals

Frequently Asked Questions

Why does

(23.10)

produce a local ring rather than a division ring?

Because the radical is no longer zero. In Wedderburn–Artin the ring equals its own semisimple quotient and the coefficient ring is $End$ of a simple module, a division ring. Here the coefficient ring is $End$ of a principal indecomposable, whose endomorphism ring is local rather than a division ring — exactly the corner $e_{1} R e_{1}$ .

k

(23.10)

canonically attached to

R

Up to isomorphism, yes; canonically as a subring, no. $k ≅ e_{1} R e_{1}$ depends on the choice of $e_{1}$ , and different choices are conjugate by a unit. What is intrinsic is the isomorphism class, pinned down by Krull–Schmidt applied to $R_{R}$ .

Why can't

(23.10)

be extended to arbitrary semiperfect rings?

Because the product decomposition of $\bar{R}$ is carried by central idempotents, and semiperfectness only guarantees that idempotents lift. $T_{2} (k)$ has $\bar{R} ≅ k \times k$ yet is indecomposable, since its centre is just the scalars. Recovering a product decomposition requires the block theory of §25.

Does

(23.11)

say a commutative semiperfect ring is artinian?

No. It says it is a finite product of local rings, and local rings need not be artinian: $k [[x]]$ and $ℤ_{p}$ are local, semiperfect and not artinian. Artinianness is the extra condition in $(23.12)$ .

Where is commutativity actually used in

(23.11)

In exactly one place: it makes the local idempotents $e_{i}$ central, so that the module decomposition $R = ⨁ e_{i} R$ is a ring decomposition. Without it, $e_{i} R$ is only a right ideal and there is no product.

How is Akizuki–Cohen related to the Hopkins–Levitzki theorem?

Hopkins–Levitzki supplies the implication artinian $\Rightarrow$ noetherian in general, and the commutative case was obtained earlier by Akizuki. The hard direction of $(23.12)$ is the converse, noetherian plus dimension zero $\Rightarrow$ artinian, proved by the filtration by products of maximal ideals.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §23, results (23.10)–(23.12).
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969, chapter on artinian rings.
H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (1960), 466–488.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964, chapters on the structure of rings with radical.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley, 1981, chapters on local rings, orders and blocks.

AI Suggested Questions

Prove that $e R ≅ f R$ for idempotents $e, f$ if and only if the images are isomorphic modulo the radical.
Give an indecomposable semiperfect ring, other than a triangular matrix ring, whose radical quotient is not simple.
For which local rings $k$ is $M_{n} (k)$ left noetherian but not left artinian?
How are the blocks of a semiperfect ring defined, and when do they correspond to the Wedderburn factors of the radical quotient?
Prove that in a noetherian commutative ring every ideal contains a finite product of prime ideals.
State and prove the noncommutative analogue of Akizuki–Cohen mentioned in the remark.
Classify the commutative semiperfect rings of finite cardinality.

Special Semiperfect Rings

Executive Summary

Overview

Learning Objectives

Definitions

Key Results

Proof Techniques and Method

Count isomorphism classes downstairs

Detect isomorphism modulo the radical

Use commutativity to centralise

Worked Example

A matrix ring over a local ring

The commutative case: $ℤ / 12$

Indecomposable with a decomposable quotient

Comparison and Classification

Relationship Map

Applications and Industry Use

Zero-dimensional rings

Reduction of orders

Codes over finite chain rings

Splitting a finite ring

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Key Results

06Proof Techniques and Method

Count isomorphism classes downstairs

Detect isomorphism modulo the radical

Use commutativity to centralise

07Worked Example

A matrix ring over a local ring

The commutative case: ℤ/12

Indecomposable with a decomposable quotient

08Comparison and Classification

09Relationship Map

10Applications and Industry Use

Zero-dimensional rings

Reduction of orders

Codes over finite chain rings

Splitting a finite ring

11Failure Modes and Common Mistakes

12Historical Notes and Lessons Learned

13Quick Reference

14Frequently Asked Questions

15Related KEVOS Topics

16References

17AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Key Results

Proof Techniques and Method

Worked Example

The commutative case: $ℤ / 12$

Comparison and Classification

Relationship Map

Applications and Industry Use

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions