The Block Decomposition of a Ring

Q: Why is the relation e e' not already an equivalence relation?

It is reflexive and symmetric, but transitivity would require the witnesses to match up, and they need not. Concretely, eR and e'R may share a composition factor, and e'R and e''R may share a different one, with eR and e''R sharing none. Blocks are the connected components, not the neighbourhoods, of this graph.

Q: Do I need artinian hypotheses to get blocks from primitive idempotents?

No. (22.5) needs only that 1 be a finite sum of orthogonal primitive idempotents, which holds for every semiperfect ring. Right artinian is a convenient sufficient condition, and it is what makes the composition-factor reformulation (22.6) available.

Q: How do blocks relate to the Wedderburn components of R/rad R?

Each block contains at least one simple component, so the number of blocks is at most the number of simple components, and each block is a union of simple components after passing to the radical quotient. Equality holds exactly when no radical element links two distinct simple components — for example when R is semisimple.

Q: Are two primitive idempotents in the same block necessarily isomorphic?

No, and this is the point of using the transitive closure. In T_2(k) the idempotents E_11 and E_22 lie in the same block but E_11R has dimension 2 and E_22R has dimension 1, so they are not isomorphic.

Q: Can a block have infinitely many primitive idempotents?

Yes. Even in M_2(k) for infinite k there are infinitely many rank-one idempotents, all primitive and all in the single block. What (22.5) makes finite is the number of blocks, not the number of idempotents.

Q: Is the block decomposition unique?

Yes, once it exists: (22.1) shows the centrally primitive idempotents are uniquely determined, so the factors R_i are determined up to order. This is what allows the linkage computation to use any convenient decomposition of the identity.

Executive Summary

A block decomposition is the finest way to write $R$ as a direct product of rings. The problem is that its defining data — centrally primitive idempotents — live in the centre, which is often invisible. §22 solves this by computing the blocks from the primitive idempotents instead, which are visible: decompose $1$ into orthogonal primitive idempotents, join two of them whenever $e R f \neq 0 \neq e' R f$ for a common $f$ , take connected components, and sum each component.

The theorem $(22.5)$ says those component sums are precisely the centrally primitive idempotents. For a right artinian ring $(22.6)$ makes the rule concrete: $e$ and $e'$ are joined exactly when $e R$ and $e' R$ share a composition factor. Blocks are then the connected components of a finite graph on the isomorphism classes of simple modules.

$e R f \neq 0$ The edge condition

ComponentsWhat blocks are

$r \leq n$ Blocks vs simple components of

R / rad R

$(22.5)$ Existence from primitive idempotents

Overview

Fix a ring $R \neq 0$ and let $E$ denote its set of primitive idempotents. The bridge between idempotents and modules is the natural isomorphism of abelian groups

{Hom}_{R} (f R, e R) ≅ e R f, a ⟼ (x \mapsto a x),

(21.6)

valid for any pair of idempotents $e, f$ of $R$ ; the corner $e R f$ is a Peirce component.

So $e R f \neq 0$ says there is a nonzero homomorphism from the projective right module $f R$ into $e R$ — a statement about how the indecomposable projectives overlap. Declaring $e \sim e'$ when some $f \in E$ receives nonzero maps into both $e R$ and $e' R$ is therefore a statement that $e R$ and $e' R$ are not independent of each other.

The relation $\sim$ is reflexive and symmetric but not transitive; the equivalence relation it generates is called linkage and written $\approx$ . The whole of $(22.5)$ is the assertion that linkage classes and blocks are the same partition, once $1$ is a finite sum of orthogonal primitive idempotents. That hypothesis is automatic for right artinian rings and, more generally, for semiperfect rings.

Learning Objectives

State the relation $e \sim e'$ on primitive idempotents and explain its module-theoretic meaning through ${Hom}_{R} (f R, e R) ≅ e R f$ .
Prove that a primitive idempotent lies in $c R$ or in $(1 - c) R$ for every central idempotent $c$ .
Prove $(22.3)$ : linked primitive idempotents lie in the same central summand.
Reconstruct the proof of $(22.5)$ that linkage class sums are centrally primitive.
Restate linkage for right artinian rings as sharing a composition factor $(22.6)$ .
Compute the linkage graph, the blocks and the block idempotents of a small algebra.

Definitions

Definition(22.2c)The linkage relation

Let $R \neq 0$ and let $E$ be its set of primitive idempotents. For $e, e' \in E$ write $e \sim e'$ if there exists $f \in E$ with $e R f \neq 0$ and $e' R f \neq 0$ . The relation $\sim$ is reflexive (take $f = e$ , and $e \in e R e$ ) and symmetric, but in general not transitive. The equivalence relation it generates is denoted $\approx$ ; explicitly $e \approx e'$ means there are $e_{1}, \dots, e_{m} \in E$ with

e \sim e_{1} \sim e_{2} \sim \dots \sim e_{m} \sim e' .

(22.2d)

We say $e$ and $e'$ are linked when $e \approx e'$ .

$E$: The set of all primitive idempotents of $R$ , not just those in some chosen decomposition of $1$ .
$e R f$: A Peirce corner; isomorphic as an abelian group to ${Hom}_{R} (f R, e R)$ , and to ${Hom}_{R} (R e, R f)$ on the other side.
Isomorphic idempotents: $e ≅ e'$ means $e R ≅ e' R$ as right $R$ -modules, equivalently $e = a b$ and $e' = b a$ for some $a, b \in R$ . Isomorphic primitive idempotents are always linked.
$rad R$: The Jacobson radical. For $f$ a local idempotent, $f R / f rad R$ is the unique simple quotient of $f R$ .
Local idempotent: An idempotent $f$ with $f R f$ a local ring. Over a semiperfect ring, primitive and local coincide.

Modules are right modules and $R$ has an identity. Nothing forces $E$ to be finite; the hypothesis of $(22.5)$ is what makes the theory finite.

Core Concepts

Two easy sources of edges

If $e, f \in E$ and $e R f \neq 0$ , then $e \sim f$ : take $f$ itself as the witness, since $f \in f R f$ is nonzero.
If $e, e' \in E$ are isomorphic idempotents, then $e \sim e'$ : take $f = e'$ , note $e' R e' \neq 0$ , and $e R e' ≅ {Hom}_{R} (e' R, e R) ≅ {Hom}_{R} (e' R, e' R) \neq 0$ .

So the linkage graph contains the isomorphism classes as cliques, and every nonzero Peirce corner as an edge. Its connected components are what we are after.

Primitive idempotents cannot straddle a central splitting

The mechanism behind everything on this page is a one-line observation. Let $c$ be a central idempotent and $e \in E$ . Then $c e$ and $(1 - c) e$ are idempotents — centrality gives $(c e)^{2} = c^{2} e^{2} = c e$ — they are orthogonal, and they sum to $e$ . Primitivity of $e$ forces one of them to vanish, so

e \in c R or e \in (1 - c) R, exclusively for e \neq 0 .

(22.4)

A primitive idempotent lies entirely inside one of the two factors.

Applying $(22.4)$ to each member of an orthogonal family of central idempotents summing to $1$ shows every primitive idempotent belongs to exactly one block. What is not obvious — and is the content of $(22.3)$ — is that linked idempotents are assigned to the same block.

Why the class sums are central

Given $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ orthogonal primitive, partition ${e_{1}, \dots, e_{n}}$ into linkage classes and let $c_{1}, \dots, c_{r}$ be the class sums. If $e_{i}$ and $e_{j}$ lie in different classes then $e_{i} R e_{j} = 0$ : a nonzero corner would give $e_{i} \sim e_{j}$ . Hence $c_{i} R c_{j} = 0$ for $i \neq j$ , and for any $a \in R$

c_{i} a = c_{i} a (\sum_{j} c_{j}) = c_{i} a c_{i} = (\sum_{j} c_{j}) a c_{i} = a c_{i} .

(22.5a)

Killing the off-diagonal corners is exactly what makes a sum of idempotents central.

Key Results

Lemma(22.3)Linkage respects central splittings

Let $R \neq 0$ be a ring, $e, e' \in E$ linked primitive idempotents, and $c$ a central idempotent of $R$ . Then $e \in c R$ if and only if $e' \in c R$ .

Proof

By symmetry and induction along a chain $(22.2 d)$ , it suffices to treat the case $e \sim e'$ , and to prove one implication. Fix $f \in E$ with $e R f \neq 0 \neq e' R f$ , and suppose $e \in c R$ , i.e. $c e = e$ .

Since $c$ is central, $0 \neq e R f = (c e) R f = e R (c f)$ . Hence $c f \neq 0$ , so by $(22.4)$ applied to $f$ we get $f \in c R$ , that is $c f = f$ .

Now $0 \neq e' R f = e' R (c f) = c (e' R f)$ , so $c e' \neq 0$ . Applying $(22.4)$ to $e'$ gives $e' \in c R$ , as required.

Theorem(22.5)Blocks from primitive idempotents

Let $R$ be a ring in which $1 = e_{1} + \dots + e_{n}$ for some pairwise orthogonal primitive idempotents $e_{1}, \dots, e_{n}$ . Then $1$ is a sum of pairwise orthogonal centrally primitive idempotents, so $R$ has a block decomposition. Moreover two primitive idempotents $e, e' \in E$ are linked if and only if they lie in the same block.

Proof

Construction. The $e_{i}$ are distinct elements of $E$ ; partition them into $\approx$ -classes and let $c_{1}, \dots, c_{r}$ be the class sums. Each $c_{i}$ is an idempotent, being a sum of orthogonal idempotents, the $c_{i}$ are pairwise orthogonal, and $c_{1} + \dots + c_{r} = 1$ . They are central by the computation $(22.5 a)$ , which uses only that $e_{i} R e_{j} = 0$ whenever $e_{i}$ and $e_{j}$ lie in different classes.

Central primitivity. Fix $i$ and write $c_{i} = e_{i_{1}} + \dots + e_{i_{m}}$ for its class. Let $c$ be a nonzero central idempotent of the ring $c_{i} R$ ; since $c_{i}$ is central in $R$ , $c$ is also central in $R$ . From $0 \neq c = c c_{i} = c e_{i_{1}} + \dots + c e_{i_{m}}$ some $c e_{i_{j}} \neq 0$ , so $e_{i_{j}} \in c R$ by $(22.4)$ . All the $e_{i_{k}}$ are linked to $e_{i_{j}}$ , so $(22.3)$ places every $e_{i_{k}}$ in $c R$ . Summing, $c_{i} \in c R$ , hence $c_{i} = c c_{i} = c$ . Thus $c_{i} R$ has no central idempotents besides $0$ and $c_{i}$ : it is indecomposable, and $c_{i}$ is centrally primitive.

Linkage classes are blocks. Let $e \in E$ be arbitrary. By $(22.4)$ applied to $c_{1}, \dots, c_{r}$ in turn, $e$ lies in exactly one block $R_{i} : = c_{i} R$ , so $e = e c_{i} \neq 0$ . Then

0 \neq e R c_{i} = e R e_{i_{1}} + \dots + e R e_{i_{m}},

(22.5b)

so $e R e_{i_{j}} \neq 0$ for some $j$ , whence $e \sim e_{i_{j}}$ and $e$ is linked to the whole class defining $c_{i}$ . Conversely, if $e'$ is linked to $e_{i_{j}}$ then $(22.3)$ puts $e'$ in $c_{i} R$ too. So the primitive idempotents lying in the block $R_{i}$ form exactly one linkage class.

Theorem(22.6)Blocks of a right artinian ring

Let $R$ be a right artinian ring and $J = rad R$ . Then $R$ has a unique block decomposition $R = R_{1} \oplus \dots \oplus R_{r}$ . For primitive idempotents $e, e' \in E$ :

$e \sim e'$ if and only if the right modules $e R$ and $e' R$ have a common composition factor;
$e$ and $e'$ lie in the same block if and only if there exist $e_{1}, \dots, e_{m} \in E$ with $e_{1} = e$ , $e_{m} = e'$ , such that $e_{i} R$ and $e_{i + 1} R$ have a common composition factor for every $i < m$ .

Proof

$R_{R}$ has finite length, so the Krull–Schmidt theorem applies and $R_{R}$ is a finite direct sum of indecomposable right ideals; this is exactly a decomposition $1 = e_{1} + \dots + e_{n}$ into orthogonal primitive idempotents. Hence $(22.5)$ gives a block decomposition, unique by $(22.1)$ .

For (1), recall that over a right artinian ring every $f \in E$ is a local idempotent, so $f R$ has $f J$ as its unique maximal submodule and $f R / f J$ is simple; conversely every simple right $R$ -module arises this way, by lifting an idempotent from the semisimple ring $R / J$ . The key computation is that for a module $M$ of finite length, ${Hom}_{R} (f R, M) ≅ M f$ , and $M \mapsto M f$ is exact. Therefore $M f \neq 0$ if and only if $S f \neq 0$ for some composition factor $S$ of $M$ ; and for a simple $S$ , $S f \neq 0$ means there is a nonzero — hence surjective — map $f R \to S$ , which forces $S ≅ f R / f J$ .

Taking $M = e R$ : the corner $e R f$ is nonzero if and only if $f R / f J$ is a composition factor of $e R$ . So $e \sim e'$ , which asks for a single $f$ with $e R f \neq 0 \neq e' R f$ , says precisely that $e R$ and $e' R$ share the composition factor $f R / f J$ . Since every simple module has this form, (1) follows, and (2) is (1) combined with the last conclusion of $(22.5)$ .

Corollary(22.6a)The semisimple case

If $R$ is semisimple, then $J = 0$ , so $e R$ is simple for every $e \in E$ and "common composition factor" means "isomorphic". Hence $\sim$ is already transitive and coincides with isomorphism of idempotents, and the blocks of $R$ are its simple components — the Wedderburn–Artin factors $M_{n_{i}} (D_{i})$ .

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Slide a central element across a corner

The identity $e R (c f) = c (e R f) = (c e) R f$ for central $c$ lets a nonzero corner transport the information $c e \neq 0$ to $c f \neq 0$ and back. This is the whole proof of $(22.3)$ .

Move 2

Primitivity as a dichotomy

Any decomposition $e = a + b$ into orthogonal idempotents must be trivial. Feeding in $a = c e$ , $b = (1 - c) e$ turns centrality of $c$ into an either-or, which is how idempotents get sorted into blocks.

Move 3

Corner vanishing gives centrality

A sum $c$ of orthogonal idempotents is central as soon as $c R (1 - c) = 0 = (1 - c) R c$ . In practice one shows all cross corners $e_{i} R e_{j}$ vanish, which is a finite check.

There is a fourth, invisible move: everything is done with a fixed decomposition of $1$ , but the conclusion — the set of blocks — does not depend on it, because $(22.1)$ makes the centrally primitive idempotents unique. This licence to compute with any convenient decomposition of $1$ is what makes the theory usable.

Worked Example

A seven-dimensional algebra with two blocks

Let $k$ be a field and let $R = T_{2} (k) \times M_{2} (k)$ , where $T_{2} (k)$ denotes the upper triangular $2 \times 2$ matrices. Then $\dim_{k} R = 3 + 4 = 7$ and $R$ is artinian. Write $e_{1} = (E_{11}, 0)$ , $e_{2} = (E_{22}, 0)$ , $e_{3} = (0, E_{11})$ , $e_{4} = (0, E_{22})$ , four orthogonal primitive idempotents with $e_{1} + e_{2} + e_{3} + e_{4} = 1$ , so $(22.5)$ applies.

Step 1: the corners

e_{1} R e_{2} = E_{11} T_{2} (k) E_{22} = k E_{12} \neq 0, e_{3} R e_{4} = E_{11} M_{2} (k) E_{22} = k E_{12} \neq 0,

(E.1)

while $e_{i} R e_{j} = 0$ whenever $i \in {1, 2}$ and $j \in {3, 4}$ , since the two factors annihilate each other. Moreover any $f \in E$ lies in one factor, so no witness can link across. The linkage graph therefore has two components, ${e_{1}, e_{2}}$ and ${e_{3}, e_{4}}$ .

Step 2: the block idempotents

The class sums are $c_{1} = e_{1} + e_{2} = (I_{2}, 0)$ and $c_{2} = e_{3} + e_{4} = (0, I_{2})$ , giving the blocks $R_{1} = T_{2} (k)$ and $R_{2} = M_{2} (k)$ . Both are indecomposable, since $Z (T_{2} (k)) = k$ and $Z (M_{2} (k)) = k$ are fields. So $r = 2$ .

Step 3: cross-check with composition factors

$R$ has three simple right modules: $S_{1}$ and $S_{2}$ , both one-dimensional, coming from the two diagonal entries of $T_{2} (k)$ , and $V = k^{2}$ coming from $M_{2} (k)$ . Then $e_{1} R$ is two-dimensional with composition factors $S_{1}, S_{2}$ , while $e_{2} R ≅ S_{2}$ . They share $S_{2}$ , so $e_{1} \sim e_{2}$ , matching $(E . 1)$ . Similarly $e_{3} R ≅ e_{4} R ≅ V$ , so $e_{3} \sim e_{4}$ . No module in the first list shares a factor with $V$ .

Process and Workflow

Decompose the identityWrite

1 = e_{1} + \dots + e_{n}

with the

e_{i}

orthogonal primitive. For a right artinian or semiperfect ring this always exists; without such a hypothesis, stop and use

(22.2)

instead.

Build the linkage graphVertices

e_{1}, \dots, e_{n}

. Draw an edge whenever

e_{i} R e_{j} \neq 0

, or — for artinian

R

— whenever

e_{i} R

and

e_{j} R

share a composition factor.

Take connected componentsTransitive closure of the edge relation. This step is where

\sim

becomes

\approx

Sum each componentThe sum

c_{i}

of the idempotents in one component is a centrally primitive idempotent;

R_{i} = c_{i} R

is a block.

Verify centralityOptional but cheap: check

c_{i} R c_{j} = 0

for

i \neq j

. If it fails, the graph was built with a missing edge.

Which hypothesis do you have?

$1$ is a finite sum of orthogonal primitive idempotentsUse

(22.5)

. This covers semiperfect rings, right artinian rings, and finite-dimensional algebras.

Only a chain condition on two-sided idealsUse

(22.2)

. It gives existence and uniqueness of the block decomposition but says nothing about linkage.

NeitherNo block decomposition need exist. An infinite product of fields is the standard warning; work with individual central idempotents instead.

Comparison and Classification

Three relations on primitive idempotents
	Isomorphism $e ≅ e'$	One step $e \sim e'$	Linkage $e \approx e'$
Reflexive	yes	yes	yes
Symmetric	yes	yes	yes
Transitive	yes	no	yes
Implies the next	yes	yes	—
Classes for semisimple $R$	simple components	simple components	simple components
Classes in general (artinian)	isomorphism types of projectives	not a partition	blocks

Three relations on primitive idempotents

Blocks versus Wedderburn components for right artinian rings
Ring	Simple components of $R / rad R$	Blocks	Comment
Semisimple $R$	$n$	$n$	the two notions coincide
$T_{n} (k)$ , $n \geq 2$	$n$	$1$	radical glues everything into one block
$T_{2} (k) \times M_{2} (k)$	$3$	$2$	worked out above
Local artinian ring	$1$	$1$	only one simple module at all
$k G$ , $char k ∤ \| G \|$	number of simple modules	same	$k G$ semisimple by Maschke

Relationship Map

Ring $R$ assume

1

is a finite sum of orthogonal primitive idempotents

Block $R_{i} = c_{i} R$ an indecomposable ring; the module category of

R

is the product of those of the blocks

Linkage class of primitive idempotentsall

e \in E

with

e \in R_{i}

, a single

\approx

-class

Isomorphism classes of primitive idempotentsseveral per block in general; one per simple module of

R_{i}

e ≅ e'

e \sim e'

e \approx e'

same block

The last arrow is an equivalence by $(22.5)$ ; the first two are strict implications in general. Reading the chain backwards is the standard error: idempotents in the same block need not be isomorphic, and there are usually several isomorphism classes of indecomposable projectives inside one block.

Applications and Industry Use

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

Modular representation theory

Brauer's block theory

For $k G$ with $char k = p$ dividing $| G |$ , the partition of simple modules into blocks given by $(22.6)$ is the starting point of Brauer theory: defect groups, blocks of defect zero and the Brauer correspondence are all attached to the blocks produced here.

Quiver algebras

Connected quivers

A finite-dimensional algebra over a field is indecomposable exactly when its Ext quiver is connected, and its blocks correspond to the connected components. The linkage graph of $(22.6)$ is a coarse version of that quiver.

Computer algebra

Splitting before decomposing

Meataxe-style algorithms first split a module algebra into blocks, because the expensive steps — finding composition series, computing endomorphism rings — then run on smaller algebras independently and in parallel.

Homological algebra

Derived and stable categories

Blocks are the indecomposable summands of the module category, so derived equivalences, Broué's abelian defect group conjecture and stable category computations are all stated block by block.

Design Considerations

Design considerations here means the choices made when modelling a problem with these algebraic structures.

Which side. The relation $e \sim e'$ is defined with right modules but is side-neutral in effect, since the resulting central idempotents are two-sided. Computing with $R e$ instead of $e R$ gives the same blocks.
**Which decomposition of $1$ .** Any orthogonal decomposition into primitive idempotents will do; the blocks do not depend on it. Choose the one with the easiest corners, usually matrix units.
Graph or algebra. For a small algebra, computing all corners $e_{i} R e_{j}$ is fastest. For a group algebra it is usually cheaper to work in the centre and factor its semisimple quotient.
How much structure to keep. The block decomposition discards nothing: $R$ is recovered as the product of its blocks. Passing to $R / rad R$ instead discards a great deal and changes the number of components, so the two reductions are not interchangeable.

Failure Modes and Common Mistakes

Do not read $e R f \neq 0$ as an isomorphism statement: it only asserts a nonzero homomorphism $f R \to e R$ , which for artinian rings means a shared composition factor, not a shared top.
Do not expect the primitive idempotents in a block to be conjugate or isomorphic; a block typically contains several isomorphism classes.
Do not confuse the block $c_{i} R$ with the indecomposable projective $e_{i} R$ : the first is an ideal and a ring, the second is a module and usually not an ideal.

Quick Reference

Edge relation

e \sim e'

iff

\exists f \in E

with

e R f \neq 0 \neq e' R f

Linkage

\approx

, the transitive closure of

\sim

Corner identity

{Hom}_{R} (f R, e R) ≅ e R f

Dichotomy

(22.4)

e \in c R

e \in (1 - c) R

for

e

primitive,

c

central idempotent

Main theorem

(22.5)

: class sums are centrally primitive; linkage classes = blocks

Artinian form

(22.6)

e \sim e'

iff

e R

e' R

share a composition factor

Semisimple caseblocks = simple components;

\sim

equals isomorphism

Count

r

blocks

\leq n

simple components of

R / rad R

Hypotheses at a glance
Statement	Hypothesis	Conclusion
$(22.4)$	$e$ primitive, $c$ central idempotent	$e$ lies in exactly one of $c R$ , $(1 - c) R$
$(22.3)$	$e \approx e'$ in $E$ , $c$ central idempotent	$e \in c R$ iff $e' \in c R$
$(22.5)$	$1$ a finite sum of orthogonal primitive idempotents	block decomposition exists; blocks = linkage classes
$(22.6)$	$R$ right artinian	unique block decomposition; linkage = shared composition factor

Frequently Asked Questions

Why is the relation

e \sim e'

not already an equivalence relation?

It is reflexive and symmetric, but transitivity would require the witnesses to match up, and they need not. Concretely, $e R$ and $e' R$ may share a composition factor, and $e' R$ and $e'' R$ may share a different one, with $e R$ and $e'' R$ sharing none. Blocks are the connected components, not the neighbourhoods, of this graph.

Do I need artinian hypotheses to get blocks from primitive idempotents?

No. $(22.5)$ needs only that $1$ be a finite sum of orthogonal primitive idempotents, which holds for every semiperfect ring. Right artinian is a convenient sufficient condition, and it is what makes the composition-factor reformulation $(22.6)$ available.

How do blocks relate to the Wedderburn components of

R / rad R

Each block contains at least one simple component, so the number of blocks is at most the number of simple components, and each block is a union of simple components after passing to the radical quotient. Equality holds exactly when no radical element links two distinct simple components — for example when $R$ is semisimple.

Are two primitive idempotents in the same block necessarily isomorphic?

No, and this is the point of using the transitive closure. In $T_{2} (k)$ the idempotents $E_{11}$ and $E_{22}$ lie in the same block but $E_{11} R$ has dimension $2$ and $E_{22} R$ has dimension $1$ , so they are not isomorphic.

Can a block have infinitely many primitive idempotents?

Yes. Even in $M_{2} (k)$ for infinite $k$ there are infinitely many rank-one idempotents, all primitive and all in the single block. What $(22.5)$ makes finite is the number of blocks, not the number of idempotents.

Is the block decomposition unique?

Yes, once it exists: $(22.1)$ shows the centrally primitive idempotents are uniquely determined, so the factors $R_{i}$ are determined up to order. This is what allows the linkage computation to use any convenient decomposition of the identity.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §22 (pp. 338–341), results (22.3), (22.4), (22.5) and (22.6).
T. Y. Lam, A First Course in Noncommutative Rings, §21, for the Peirce corner isomorphism and local idempotents, and §19 for Krull–Schmidt.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley-Interscience, 1981, on blocks and linkage of indecomposable projectives.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §27 on semiperfect rings and idempotents.
J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics 11, Cambridge University Press, 1986, for blocks of group algebras.

AI Suggested Questions

Give an explicit finite-dimensional algebra in which the relation $e \sim e'$ fails to be transitive.
How does the linkage graph of a finite-dimensional algebra compare with its Ext quiver?
Prove that a semiperfect ring always satisfies the hypothesis of $(22.5)$ .
For $k G$ with $G$ a finite $p$ -group and $char k = p$ , show that there is exactly one block.
What are the blocks of the path algebra of a quiver, and how do they relate to its connected components?
Explain the isomorphism ${Hom}_{R} (f R, M) ≅ M f$ and why $M \mapsto M f$ is an exact functor.
How many blocks does the group algebra of a symmetric group have in characteristic $p$ , and what is the Nakayama conjecture saying about them?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Two easy sources of edges

Primitive idempotents cannot straddle a central splitting

Why the class sums are central

06Key Results

07Proof Techniques and Method

Slide a central element across a corner

Primitivity as a dichotomy

Corner vanishing gives centrality

08Worked Example

A seven-dimensional algebra with two blocks

Step 1: the corners

Step 2: the block idempotents

Step 3: cross-check with composition factors

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Brauer's block theory

Connected quivers

Splitting before decomposing

Derived and stable categories

13Design Considerations

14Failure Modes and Common Mistakes

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

Design Considerations

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions