Basic Rings and Morita Reduction

Q: Why is the basic ring unique when the basic idempotent is not?

Because eRe ≅ End_R(eR) and the module eR — one copy of each principal indecomposable — is determined up to isomorphism by R alone. Different choices of e produce isomorphic modules, hence isomorphic endomorphism rings.

Q: Does every ring have a basic ring?

The construction as given requires R to be semiperfect, since it needs a finite complete set of principal indecomposables and a decomposition of 1 into orthogonal primitive idempotents. For a general ring neither exists, and there is no substitute.

Q: If two semiperfect rings have isomorphic basic rings, are they isomorphic?

No — they are Morita equivalent, which is strictly weaker. D and M_n(D) have the same basic ring D but are not isomorphic for n ≥ 2. What is recovered from the basic ring together with the multiplicities n_1, …, n_r is the Morita class plus the choice of progenerator.

Executive Summary

Let $R$ be semiperfect and $B = e R e$ a basic ring of $R$ . Lam's $(25.8)$ shows that $B$ retains essentially everything: the right ideals of $B$ embed in those of $R$ , the ideals of $B$ and of $R$ correspond by an isomorphism of lattices that respects multiplication and matches $rad B$ with $rad R$ , and the primitive idempotents, principal indecomposables and blocks correspond one-to-one.

The definition $(25.9)$ calls $R$ basic when $1$ is itself a basic idempotent, and the criterion $(25.10)$ makes this checkable: $R$ is basic if and only if $R / rad R$ is a finite direct product of division rings. Every semiperfect ring is Morita equivalent to a basic one, and the basic ring is the canonical representative of its Morita class.

$B = e R e$ Basic ring

IdealsLattices isomorphic

$\prod_{i} D_{i}$ Criterion for basic

MoritaSame module category

Overview

Wedderburn–Artin says a semisimple ring is a product of matrix rings over division rings. Matrix sizes are invisible to module theory: $Mod - M_{n} (D)$ and $Mod - D$ are equivalent categories. So a semisimple ring can always be replaced, for categorical purposes, by a product of division rings.

The basic ring construction performs the same reduction for semiperfect rings, where there is a radical in the way. The mechanism is a full idempotent and the corner ring theory of §21.

R semiperfect ⟶ B = e R e (e basic), \bar{B} ≅ D_{1} \times \dots \times D_{r},

(25.9)–(25.10)

All matrix sizes become $1$ in the semisimple quotient of the basic ring; the radical, the ideal lattice and the block structure are untouched.

Lam develops the Morita theory of module category equivalences in the sequel volume; §25 proves the concrete correspondences directly, without the general machinery, and that is what is recorded here.

Learning Objectives

State $(25.8)$ in full, distinguishing which parts need $e$ to be full.
Prove $f R f = f B f$ for an idempotent $f \in B$ and deduce that primitivity is detected inside $B$ .
Prove the criterion $(25.10)$ for a semiperfect ring to be basic.
Give examples separating basic from indecomposable in both directions.
Compute the basic ring of a block triangular matrix ring and check the invariants match.
Explain in what sense the basic ring is the canonical representative of a Morita class.

Definitions

Definition(25.9)Basic ring

A semiperfect ring $R$ is called basic if $1$ is a basic idempotent of $R$ — equivalently, if in some (hence any) decomposition $1 = e_{1} + \dots + e_{n}$ into orthogonal primitive idempotents the modules $e_{1} R, \dots, e_{n} R$ are pairwise non-isomorphic. Equivalently again, $R$ is its own basic ring.

$B = e R e$: A basic ring of $R$ , for $e$ a basic idempotent; identity element $e$ .
Ideal lattice map: $𝔄 \mapsto R 𝔄 R$ from ideals of $B$ to ideals of $R$ , with inverse $𝔅 \mapsto e 𝔅 e$ .
$E$ , $E_{0}$: The sets of primitive idempotents of $R$ and of $B$ respectively; the theorem gives $E_{0} = E \cap B$ .
Morita equivalence: An equivalence of module categories. For a full idempotent, $M \mapsto M e$ implements an equivalence between right $R$ -modules and right $e R e$ -modules.

Core Concepts

Corners see their own idempotents

The technical heart of $(25.8)$ is an identity so short it is easy to miss. If $f$ is an idempotent of $B = e R e$ , then $e f = f e = f$ , hence

f R f = (f e) R (e f) = f (e R e) f = f B f .

(25.8)(3)

The corner of $R$ at $f$ and the corner of $B$ at $f$ are the same ring. Locality, primitivity and all local-idempotent phenomena therefore transfer verbatim.

Since both $R$ and $B$ are semiperfect, primitive is the same as local for both, so $f$ is primitive in $B$ exactly when it is primitive in $R$ : $E_{0} = E \cap B$ .

Fullness and ideals

For an arbitrary idempotent $e$ , the corner ring theory gives an injective inclusion-preserving map from right ideals of $e R e$ to right ideals of $R$ , and $e (R 𝔄 R) e = 𝔄$ for ideals $𝔄$ of $e R e$ . Fullness upgrades the second map to a surjection, hence an isomorphism of ideal lattices — and because a basic idempotent is always full, the upgrade is automatic here.

e

basic

e

fullideals correspondradicals and blocks correspond

Why the criterion looks the way it does

Being basic is a statement about multiplicities, and multiplicities are visible in $\bar{R}$ : the module $e_{i} R$ occurs $n_{i}$ times in $R_{R}$ , where $n_{i}$ is the size of the $i$ -th matrix factor of $\bar{R}$ . All multiplicities equal $1$ precisely when every matrix factor is $1 \times 1$ , i.e. when $\bar{R}$ is a product of division rings. That is the whole of $(25.10)$ .

Key Results

Theorem(25.8)What a basic ring retains

Let $R$ be a semiperfect ring, $e$ a basic idempotent, and $B = e R e$ the corresponding basic ring. Then:

$𝔄 \mapsto 𝔄 R$ is an injective, inclusion-preserving map from the lattice of right ideals of $B$ into that of $R$ ; this part holds for any idempotent $e$ .
$𝔄 \mapsto R 𝔄 R$ is an isomorphism from the lattice of two-sided ideals of $B$ onto that of $R$ , with inverse $𝔅 \mapsto e 𝔅 e$ ; it respects products of ideals and carries $rad B$ to $rad R$ .
$E_{0} = E \cap B$ , where $E$ and $E_{0}$ are the primitive idempotents of $R$ and of $B$ ; moreover for $f, f' \in E_{0}$ , isomorphism of $f$ and $f'$ holds in $B$ if and only if it holds in $R$ , and the same is true of the relation $\sim$ and of linkage.
The map in (1) induces a bijection between isomorphism types of principal indecomposable right $B$ -modules and those of right $R$ -modules; the map in (2) induces a bijection between the blocks of $B$ and the blocks of $R$ .

Proof

Parts (1) and (2) are the corner ring statements $(21.11)$ : (1) needs nothing beyond $e$ idempotent, while surjectivity in (2) needs $R e R = R$ , which holds because a basic idempotent is full by $(25.6)$ . That the correspondence preserves products of ideals is part of $(21.11) (2)$ , and applying it to the unique largest ideal with semisimple quotient identifies $rad B$ with $rad R$ .

(3). For an idempotent $f \in B$ we have $e f = f e = f$ , so $f R f = f (e R e) f = f B f$ . Hence $f$ is a local idempotent of $R$ if and only if it is one of $B$ ; since $R$ and $B$ are semiperfect, local and primitive coincide, giving $E_{0} = E \cap B$ .

If $f ≅ f'$ in $B$ then trivially $f ≅ f'$ in $R$ . Conversely, if $f ≅ f'$ in $R$ , then by $(21.20)$ there are $a \in f R f'$ and $b \in f' R f$ with $a b = f$ and $b a = f'$ . But $a = f a f' \in (f e) R (e f') = f B f' \subseteq B$ and likewise $b \in B$ , so the same equations witness $f ≅ f'$ inside $B$ .

For linkage, suppose $f \sim f'$ in $R$ , say $f R g \neq 0 \neq f' R g$ with $g \in E$ . By the definition of a basic idempotent there is $g_{0} \in E_{0}$ with $g_{0} R ≅ g R$ , and then $f R g ≅ {Hom}_{R} (g R, f R) ≅ {Hom}_{R} (g_{0} R, f R) ≅ f R g_{0}$ , so $f R g_{0} \neq 0$ ; similarly $f' R g_{0} \neq 0$ . Since $f R g_{0} = f B g_{0}$ and $f' R g_{0} = f' B g_{0}$ , we get $f \sim f'$ in $B$ . The converse is immediate, and linkage, being the generated equivalence relation, transfers with $\sim$ .

(4). For $f, f' \in E_{0}$ we have $f B ≅ f' B$ if and only if $f ≅ f'$ in $B$ , if and only if $f ≅ f'$ in $R$ , if and only if $f R ≅ f' R$ ; and every principal indecomposable of $R$ is isomorphic to $g_{0} R$ for some $g_{0} \in E_{0}$ , by the definition of a basic idempotent. This is the asserted bijection. For blocks, the lattice isomorphism of (2) matches direct sum decompositions of $B$ into indecomposable ideals with those of $R$ ; equivalently, by (3) the linkage classes of $E_{0}$ and of $E$ correspond, and by $(25.4)$ linkage classes are blocks.

Proposition(25.10)Criterion for being basic

A semiperfect ring $R$ is basic if and only if $\bar{R} = R / rad R$ is a finite direct product of division rings.

Proof

Take $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ orthogonal primitive, hence local, idempotents. Reduction modulo $J = rad R$ gives an orthogonal decomposition $\bar{1} = {\bar{e}}_{1} + \dots + {\bar{e}}_{n}$ into primitive idempotents of $\bar{R}$ , and by $(19.27)$ , $e_{i} R ≅ e_{j} R$ if and only if ${\bar{e}}_{i} \bar{R} ≅ {\bar{e}}_{j} \bar{R}$ . So $1$ is a basic idempotent of $R$ if and only if $\bar{1}$ is a basic idempotent of $\bar{R}$ .

Now $\bar{R}$ is semisimple, say $\bar{R} ≅ \prod_{i = 1}^{r} M_{n_{i}} (D_{i})$ . A decomposition of $\bar{1}$ into orthogonal primitive idempotents corresponds to a decomposition of $\bar{R}$ into minimal right ideals, in which the simple module belonging to the $i$ -th factor occurs exactly $n_{i}$ times. These are pairwise non-isomorphic precisely when every $n_{i} = 1$ , that is, when $\bar{R} ≅ D_{1} \times \dots \times D_{r}$ .

Corollary—Basic versus indecomposable

A semiperfect ring $R$ is indecomposable if and only if one — equivalently every — basic ring of $R$ is indecomposable. No implication holds between basic and indecomposable in either direction: $D_{1} \times \dots \times D_{t}$ is basic and decomposable for $t \geq 2$ , while $M_{n} (D)$ is indecomposable and not basic for $n \geq 2$ .

Proof

By $(25.8) (2)$ the ideal lattices of $R$ and $B$ are isomorphic, so $R$ is a direct sum of two nonzero ideals exactly when $B$ is; equivalently, by $(25.8) (4)$ , they have the same number of blocks. The two examples are immediate from $(25.10)$ : $D_{1} \times \dots \times D_{t}$ equals its own semisimple quotient, and $M_{n} (D)$ has semisimple quotient $M_{n} (D)$ , not a product of division rings when $n \geq 2$ .

Remark—Morita interpretation

Because $e$ is full, $M \mapsto M e$ is an equivalence from right $R$ -modules to right $B$ -modules, so $B$ is Morita equivalent to $R$ . Every semiperfect ring is therefore Morita equivalent to a basic one, and among the rings in a Morita class the basic one is unique up to isomorphism. This is the sense in which $B$ is a canonical representative; the general theory is developed in Lam's sequel volume.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Absorb the idempotent

For $f \in e R e$ one has $e f = f e = f$ , so expressions like $f R f'$ automatically live inside $B$ . Most of $(25.8) (3)$ is this observation applied three times.

Move 2

Replace a witness by a representative

A witness idempotent $g$ outside $B$ can be swapped for $g_{0} \in E_{0}$ with $g_{0} R ≅ g R$ , because $f R g ≅ {Hom}_{R} (g R, f R)$ depends only on the isomorphism type of $g R$ .

Move 3

Transport structure along a lattice isomorphism

Once ideals correspond and products are respected, everything defined lattice-theoretically — radical, blocks, nilpotency, primeness of ideals — corresponds automatically. No further computation is required.

Move 3 explains the shape of $(25.8)$ : rather than proving each invariant separately, one proves a single structural correspondence and then reads off the consequences. The price is that the correspondence must respect multiplication, which is where fullness is genuinely needed.

Worked Example

A block triangular ring and its basic ring

Let $k$ be a division ring and let $R \subseteq M_{3} (k)$ consist of the block upper triangular matrices with blocks of sizes $2$ and $1$ :

R = {(\begin{matrix} a_{11} & a_{12} & b_{1} \\ a_{21} & a_{22} & b_{2} \\ 0 & 0 & c \end{matrix}) : a_{i j}, b_{i}, c \in k}, \dim_{k} R = 7 .

(E.1)

Its radical is the block of $b$ 's, $J = rad R ≅ k^{2}$ with $J^{2} = 0$ , and $\bar{R} ≅ M_{2} (k) \times k$ . By $(25.10)$ , $R$ is not basic. The diagonal matrix units $e_{1} = E_{11}$ , $e_{2} = E_{22}$ , $e_{3} = E_{33}$ are orthogonal local idempotents summing to $1$ , with $e_{1} R ≅ e_{2} R$ (the first two rows) and $e_{3} R$ of $k$ -dimension $1$ .

So $r = 2$ and $e = e_{1} + e_{3}$ is a basic idempotent. Computing the corner:

B = e R e = {(\begin{matrix} a_{11} & b_{1} \\ 0 & c \end{matrix})} ≅ T_{2} (k), \dim_{k} B = 3 .

(E.2)

The $(3, 1)$ entry of any element of $R$ is zero, so the corner is upper triangular. The basic ring of $R$ is $T_{2} (k)$ .

Checking the correspondences

Principal indecomposables. $R$ has two types, $e_{1} R$ (dimension $3$ ) and $e_{3} R$ (dimension $1$ ); $B$ has two, of dimensions $2$ and $1$ . The count matches; the dimensions need not.
Simple modules. $R$ has $V_{1}$ of dimension $2$ and $V_{2}$ of dimension $1$ ; $B$ has two one-dimensional simples. Again only the count is an invariant.
Cartan matrix. $e_{1} R$ has composition factors $V_{1}$ then $V_{2}$ , and $e_{3} R = V_{2}$ , so $C = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ — exactly the Cartan matrix of $T_{2} (k)$ .
Blocks. $e_{1} R e_{3} = k E_{13} \neq 0$ , so $e_{1}$ and $e_{3}$ are linked and $R$ is indecomposable; correspondingly $T_{2} (k)$ is indecomposable. Each has exactly one block.
Ideals. Both rings have exactly five two-sided ideals, arranged in the same lattice, as $(25.8) (2)$ requires.

A degenerate check

Applying the same recipe to $R = M_{3} (k)$ gives $r = 1$ , $e = E_{11}$ and $B ≅ k$ : dimension collapses from $9$ to $1$ , both rings are indecomposable, both have one simple module, and both have Cartan matrix $(1)$ .

Process and Workflow

Is my semiperfect ring $R$ basic?

Compute $\bar{R}$ If

R / rad R

is a finite product of division rings, yes; otherwise no. This is

(25.10)

and is usually the cheapest test.

Inspect the idempotentsDecompose

1

into orthogonal primitive idempotents and check whether any two give isomorphic right ideals. A repetition means not basic.

It is not basicForm a basic idempotent by keeping one representative per type and replace

R

e R e

for all categorical purposes.

ReduceCompute

rad R

and the Wedderburn decomposition of

\bar{R}

; read off

r

and the multiplicities

n_{i}

ChoosePick one primitive idempotent per type and sum them to get a basic idempotent

e

CornerForm

B = e R e

and verify

\bar{B}

is a product of division rings.

TransportTranslate the original question into

B

: ideals, blocks, simple modules, Cartan matrix and module category all carry across.

Comparison and Classification

Basic and indecomposable are independent
Ring	Basic?	Indecomposable?	Basic ring
Division ring $D$	yes	yes	$D$
$M_{n} (D)$ , $n \geq 2$	no	yes	$D$
$D_{1} \times \dots \times D_{t}$ , $t \geq 2$	yes	no	itself
$T_{n} (D)$ , $n \geq 2$	yes	yes	itself
$M_{2} (D) \times M_{3} (D)$	no	no	$D \times D$
Local ring $k$	yes	yes	itself
$ℤ / 12 ℤ$	yes	no	itself
Block triangular ring of $(E . 1)$	no	yes	$T_{2} (k)$

What passes to the basic ring
Invariant	Preserved?	Reason
Lattice of two-sided ideals	yes	(25.8)(2)
Product of ideals, radical	yes	(25.8)(2)
Number of simple modules	yes	(25.8)(4)
Number of blocks	yes	(25.8)(4)
Cartan matrix	yes	Morita invariance of composition multiplicities
Module category	yes	$M \mapsto M e$ is an equivalence
Dimension over a field	no	$9 \to 1$ for $M_{3} (k)$
Matrix sizes $n_{i}$ in $\bar{R}$	no	all become $1$ by $(25.10)$
Centre as a subring of $R$	no	the identities differ; centres are isomorphic as rings but not as subrings

Relationship Map

Semiperfect rings

\bar{R}

semisimple and idempotents lift

Basic semiperfect rings

\bar{R}

a finite product of division rings

Basic and indecomposableone block; e.g.

T_{n} (D)

, local rings

Local rings

\bar{R}

a single division ring

R

semiperfect

B = e R e

basic

Mod - R ≃ Mod - B

same ideals, blocks, Cartan matrix

Over an algebraically closed field the chain terminates in a very concrete place: every basic finite-dimensional algebra is isomorphic to a quotient of a path algebra $k Q$ by an admissible ideal, so basic rings are the algebras one writes down by drawing a quiver.

Applications and Industry Use

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

Representation theory of algebras

Quivers and relations

Gabriel's theorem says a basic finite-dimensional algebra over an algebraically closed field is $k Q / I$ for a quiver $Q$ and admissible ideal $I$ . Reduction to the basic ring is the first step in every classification of finite-dimensional algebras by representation type.

Computer algebra

Smaller models

Replacing an algebra by its basic ring shrinks the dimension without changing the module category, which reduces the cost of module-theoretic computations; systems that decompose modules typically work with the basic algebra internally.

Modular representation theory

Basic algebras of blocks

Comparing two block algebras up to Morita equivalence means comparing their basic algebras. Donovan's conjecture and the classification of blocks with a given defect group are stated in exactly these terms.

Integral representation theory

Orders and lattices

An order over a complete discrete valuation ring is semiperfect, so it has a basic order with the same lattice category; classifications of integral representations proceed via basic orders.

The consistent theme is normalisation: the basic ring is what one computes with when the answer is only supposed to depend on the module category.

Failure Modes and Common Mistakes

Do not apply $(25.10)$ to a ring that has not been shown semiperfect; the criterion presupposes it.
Do not assume a basic ring is commutative, artinian, or hereditary because a familiar example was. $T_{n} (D)$ is hereditary; most basic algebras are not.
Do not identify the centre of $B$ with the centre of $R$ inside $R$ : they are isomorphic as rings, via $z \mapsto e z e$ , but they are different subsets with different identity elements.
Do not expect the basic ring to be a subring of $R$ containing $1$ ; it is a corner, with identity $e$ .

Historical Notes and Lessons Learned

1908–1927Wedderburn and ArtinStructure theory for semisimple algebras and for rings with chain conditions establishes that matrix sizes and division rings are the only invariants in the semisimple case.
1941Brauer's blocksBrauer's work on modular characters makes block decompositions and Cartan invariants central objects, creating the need for a reduced model of a group algebra.
1940sBrauer and Osima construct basic ringsFor right artinian rings the basic ring is constructed explicitly, years before any general theory of equivalences of module categories exists.
1958Morita's theoremsMorita characterises equivalences of module categories by progenerators, explaining conceptually why the basic ring construction preserves everything module-theoretic.
1972Gabriel's quiversBasic finite-dimensional algebras over an algebraically closed field are identified with path algebras modulo admissible relations, making the basic ring a combinatorial object.

The lesson is that a construction can precede the theory that explains it. The basic ring was built because representation theorists needed a smaller model; only later did Morita theory show that what had been preserved was precisely the module category, and nothing more.

Quick Reference

Setting

R

semiperfect,

e

a basic idempotent,

B = e R e

Right ideals

𝔄 \mapsto 𝔄 R

is an injective inclusion-preserving embedding

Ideals

𝔄 \mapsto R 𝔄 R

is a lattice isomorphism, respecting products

Radical

R (rad B) R = rad R

and

rad B = e (rad R) e

Idempotents

E_{0} = E \cap B

; isomorphism and linkage agree in

B

and

R

Modules and blocksPrincipal indecomposables and blocks correspond one-to-one

Criterion

R

basic

\Leftrightarrow

R / rad R

a finite product of division rings

Morita

M \mapsto M e

is an equivalence;

B

is the canonical model of the Morita class

Statement locator
Fact	Statement	Reference
Retention theorem	ideals, idempotents, principal indecomposables, blocks	(25.8)
Definition	$R$ basic iff $1$ is a basic idempotent	(25.9)
Criterion	$\bar{R}$ a finite product of division rings	(25.10)
Fullness	basic idempotents are full	(25.6)
Corner lattices	$𝔄 \mapsto R 𝔄 R$ onto when $R e R = R$	(21.11)
Isomorphic idempotents	$e ≅ f$ iff $e R ≅ f R$	(21.20)
Lifting criterion	$P ≅ Q$ iff $P / P J ≅ Q / Q J$	(19.27)

Frequently Asked Questions

Why is the basic ring unique when the basic idempotent is not?

Because $e R e ≅ {End}_{R} (e R)$ and the module $e R$ — one copy of each principal indecomposable — is determined up to isomorphism by $R$ alone. Different choices of $e$ produce isomorphic modules, hence isomorphic endomorphism rings.

Does every ring have a basic ring?

The construction as given requires $R$ to be semiperfect, since it needs a finite complete set of principal indecomposables and a decomposition of $1$ into orthogonal primitive idempotents. For a general ring neither exists, and there is no substitute.

If two semiperfect rings have isomorphic basic rings, are they isomorphic?

No — they are Morita equivalent, which is strictly weaker. $D$ and $M_{n} (D)$ have the same basic ring $D$ but are not isomorphic for $n \geq 2$ . What is recovered from the basic ring together with the multiplicities $n_{1}, \dots, n_{r}$ is the Morita class plus the choice of progenerator.

Is the basic ring of a commutative ring commutative?

It is the ring itself. A commutative semiperfect ring is a finite product of local rings, so its semisimple quotient is a product of fields and $(25.10)$ says it is already basic. Non-commutative rings can of course have commutative basic rings, as $M_{n} (k)$ shows.

How does the Cartan matrix behave under the reduction?

It is unchanged, up to simultaneous permutation of rows and columns. Composition multiplicities of simple modules in principal indecomposables are categorical data, and $M \mapsto M e$ matches the two families of modules.

Why do people say the basic ring is the canonical representative of a Morita class?

Because every semiperfect ring is Morita equivalent to a basic one, and two basic semiperfect rings that are Morita equivalent are isomorphic. So each Morita class of semiperfect rings contains exactly one basic ring up to isomorphism.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §25 (pp. 376–378).
T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, 1999, §18 (Morita theory of category equivalences).
K. Morita, “Duality for modules and its applications to the theory of rings with minimum condition”, Science Reports of the Tokyo Kyoiku Daigaku, Section A, 6 (1958), 83–142.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §21–§22 and §27.
I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1, London Mathematical Society Student Texts 65, Cambridge University Press, 2006, Chapters I–III.

AI Suggested Questions

Prove that two Morita equivalent basic semiperfect rings are isomorphic.
Compute the basic algebra of $k S_{4}$ in characteristic $2$ and describe its quiver.
Show that the block triangular ring of the worked example is right hereditary, and decide whether it is left hereditary.
What is the basic ring of the ring of $n \times n$ matrices over a commutative local ring, and why does the answer not depend on $n$ ?
How does Gabriel's theorem fail over a field that is not algebraically closed, and what replaces the quiver there?
Describe the relationship between basic orders and maximal orders over a complete discrete valuation ring.
Which properties of a ring are Morita invariant, and which of the ones used in this collection are not?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Corners see their own idempotents

Fullness and ideals

Why the criterion looks the way it does

06Key Results

07Proof Techniques and Method

Absorb the idempotent

Replace a witness by a representative

Transport structure along a lattice isomorphism

08Worked Example

A block triangular ring and its basic ring

Checking the correspondences

A degenerate check

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Quivers and relations

Smaller models

Basic algebras of blocks

Orders and lattices

13Failure Modes and Common Mistakes

14Historical Notes and Lessons Learned

15Quick Reference

16Frequently Asked Questions

17Related KEVOS Topics

18References

19AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Worked Example

Process and Workflow

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