Principal Indecomposable Modules

Q: Why are they called *principal* indecomposables?

Because they are principal right ideals — eR is generated by the single element e — that happen to be indecomposable as modules. The name predates the general theory and comes from the classical study of right artinian rings, where these modules were first isolated.

Q: Is every indecomposable projective right module a principal indecomposable?

Every finitely generated one is, over a semiperfect ring: it is a summand of a free module of finite rank, and (25.3)(2) decomposes it into modules e_iR; indecomposability forces exactly one summand. Without finite generation this argument is unavailable and the statement should not be assumed.

Q: How do I tell whether eR ≅ fR without decomposing anything?

Use the idempotent criterion (21.20): eR ≅ fR as right modules if and only if there exist a ∈ eRf and b ∈ fRe with ab = e and ba = f. Equivalently, reduce modulo the radical and compare e,R with f,R inside the semisimple ring R.

Q: Does eR determine Re?

The idempotent e determines both, and eR ≅ fR holds if and only if Re ≅ Rf, since both are equivalent to e and f being isomorphic idempotents. But the modules themselves can look completely different: over T_n(k) the right module e_1R has length n while the left module Re_1 is simple.

Q: What replaces this theory when R is only semilocal?

Very little. Semilocal rings give a semisimple quotient R, so simple modules are still classified, but without idempotent lifting there is no way back: primitive idempotents of R need not be local, and there may be no finite orthogonal decomposition of 1 into primitive idempotents at all.

Q: How does the composition length of eR relate to the Cartan matrix?

When R is right artinian, the length of e_iR is the sum of the i-th row of the Cartan matrix, since the (i,j) entry counts the composition factors of e_iR isomorphic to V_j. The diagonal entries are always at least 1 because V_i appears as the top of e_iR.

Executive Summary

Let $R$ be a semiperfect ring with $J = rad R$ , and let $E$ be its set of primitive idempotents. The right modules $e R$ for $e \in E$ — the principal indecomposables — are exactly the building blocks of projective modules over $R$ : each is cyclic, projective, and strongly indecomposable, and each has a unique maximal submodule $e J$ .

Lam's $(25.3)$ turns this into a complete classification: $e R \mapsto e R / e J$ is a bijection from isomorphism types of principal indecomposables onto isomorphism types of simple right $R$ -modules, and every finitely generated projective right $R$ -module is a direct sum of principal indecomposables with uniquely determined multiplicities. The projective world over a semiperfect ring is therefore as rigid as the semisimple world it sits above.

$e R$ Principal indecomposable

$e R / e J$ Its unique simple top

1–1PIMs vs simple modules

(25.3)Classification theorem

Overview

For a semisimple ring the indecomposable projectives are the simple modules themselves, and Wedderburn–Artin lists them. For an arbitrary ring there is no list at all: indecomposable projectives may be wildly non-cyclic and the Krull–Schmidt property fails. Semiperfect rings are the natural halfway house, and the principal indecomposables are what survives the passage from $R / J$ back to $R$ .

Two facts from the idempotent theory do all the work. First, semiperfectness gives a decomposition

1 = e_{1} + e_{2} + \dots + e_{n}, R_{R} = e_{1} R \oplus e_{2} R \oplus \dots \oplus e_{n} R,

(23.6)

The $e_{i}$ are mutually orthogonal local idempotents; the corresponding Peirce decomposition of the regular module is into principal indecomposables.

Second, idempotents lift modulo $J$ , so nothing visible in the semisimple quotient $\bar{R} = R / J$ is lost on the way up. Between them these give a dictionary: simple right $\bar{R}$ -modules downstairs, principal indecomposables upstairs, and the two lists have the same length.

Following Lam's §25 we work with right modules throughout. Every statement has a left analogue obtained by reading Re in place of eR; the two theories are formally symmetric but the actual modules on the two sides can be very different, as the worked example shows.

Learning Objectives

Recognise the standing hypotheses: $R$ semiperfect, $J = rad R$ , $E$ the primitive idempotents.
Prove that $e R / e J$ is simple and that $e J = rad (e R)$ is the unique maximal submodule.
State the bijection of $(25.3) (1)$ and identify both of its directions.
Decompose a finitely generated projective $P$ as $n_{1} (e_{1} R) \oplus \dots \oplus n_{r} (e_{r} R)$ and read off the $n_{i}$ from $P / P J$ .
Explain why ${End}_{R} (e R) ≅ e R e$ is local and what that buys via Krull–Schmidt.
Compute all principal indecomposables of $T_{3} (k)$ for a division ring $k$ .

Definitions

Definition(25.2)Principal indecomposable module

Let $R$ be a semiperfect ring, $J = rad R$ , and let $E$ denote the set of primitive idempotents of $R$ . For $e \in E$ the right ideal $e R$ is called a principal indecomposable module. These modules are cyclic (generated by $e$ ), projective (a direct summand of $R_{R}$ , since $R = e R \oplus (1 - e) R$ ), and strongly indecomposable.

Primitive idempotent: $e = e^{2} \neq 0$ that is not a sum of two nonzero orthogonal idempotents; equivalently $e R$ is an indecomposable right module.
Local idempotent: $e = e^{2}$ with $e R e$ a local ring. Local always implies primitive; over a semiperfect ring the converse also holds, which is $(25.1)$ .
$\bar{R}$ , $\bar{a}$: $\bar{R} = R / J$ with $J = rad R$ , and $\bar{a} = a + J$ . For semiperfect $R$ the ring $\bar{R}$ is semisimple.
$rad M$: The intersection of the maximal submodules of $M$ . For $M = e R$ over a semiperfect ring, $rad (e R) = e J$ .
Strongly indecomposable: ${End}_{R} (M)$ is local. For $M = e R$ one has ${End}_{R} (e R) ≅ e R e$ , so $e$ local means exactly $e R$ strongly indecomposable.

Core Concepts

From idempotents to modules and back

The whole subject rests on one adjunction-free identity: for any idempotent $e$ and any right $R$ -module $M$ ,

{Hom}_{R} (e R, M) ≅ M e, in particular {End}_{R} (e R) ≅ e R e .

(21.6)

A homomorphism $e R \to M$ is determined by the image of $e$ , which may be any element of $M e$ .

So questions about the module $e R$ become questions about the corner ring $e R e$ , which is the subject of the Corner Rings page. Indecomposability of $e R$ corresponds to primitivity of $e$ ; strong indecomposability corresponds to $e R e$ being local.

Why the top is simple

For any idempotent $e$ there is an isomorphism of right $\bar{R}$ -modules $e R / e J ≅ \bar{e} \bar{R}$ . When $e$ is local, $\bar{e} \bar{R}$ is a minimal right ideal of the semisimple ring $\bar{R}$ , hence simple; and since $J$ annihilates it, it is simple as a right $R$ -module too. The submodule $e J$ is then maximal, and it contains every proper submodule of $e R$ — for if $𝔄 \subseteq e R$ is a submodule not inside $e J$ , then $𝔄 + e J = e R$ and Nakayama's Lemma forces $𝔄 = e R$ .

e \in E

e R e

local

e R

strongly indec.

e R / e J

simple

Projective covers

Because $e J$ is small in $e R$ (Nakayama again), the surjection $e R ↠ e R / e J$ is a projective cover. That is the conceptual reason the correspondence in $(25.3) (1)$ is a bijection rather than merely a surjection: projective covers are unique up to isomorphism, so the simple module determines $e R$ .

Key Results

Proposition(25.1)Primitive idempotents are local

Let $R$ be a semiperfect ring. Then every primitive idempotent $e \in R$ is a local idempotent, i.e. $e R e$ is a local ring.

Proposition(25.2)The simple top

Let $R$ be semiperfect, $J = rad R$ and $e \in E$ . Then $e R / e J ≅ \bar{e} \bar{R}$ is a simple right $\bar{R}$ -module, hence a simple right $R$ -module; $e J$ is the unique maximal submodule of $e R$ ; and $rad (e R) = e J$ .

Theorem(25.3)Classification of principal indecomposables

Let $R$ be a semiperfect ring with $J = rad R$ and $E$ its set of primitive idempotents.

The assignment $e R \mapsto e R / e J$ ( $e \in E$ ) induces a bijection between the isomorphism types of principal indecomposable right $R$ -modules and the isomorphism types of simple right $R$ -modules. In particular there are only finitely many of each, say $r$ .
Let $e_{1} R, \dots, e_{r} R$ be a complete irredundant list of principal indecomposables. Then every finitely generated projective right $R$ -module $P$ satisfies $P ≅ n_{1} (e_{1} R) \oplus \dots \oplus n_{r} (e_{r} R)$ for uniquely determined integers $n_{i} \geq 0$ .
For such $P$ , the ring ${End}_{R} (P)$ is semiperfect.

Proof

Well-definedness and injectivity in (1). An isomorphism $e R ≅ e' R$ carries $rad (e R) = e J$ onto $rad (e' R) = e' J$ , so it induces $e R / e J ≅ e' R / e' J$ . Conversely suppose $e R / e J ≅ e' R / e' J$ . Both $e R$ and $e' R$ are finitely generated projective, and the displayed quotients are their reductions modulo $J$ ; by the lifting lemma $(19.27)$ — for finitely generated projectives $P, Q$ one has $P ≅ Q$ if and only if $P / P J ≅ Q / Q J$ — we conclude $e R ≅ e' R$ .

Surjectivity in (1). Let $V$ be a simple right $R$ -module. Then $V J = 0$ , so $V$ is a simple right $\bar{R}$ -module. As $\bar{R}$ is semisimple, $V ≅ \bar{x} \bar{R}$ for some right irreducible idempotent $\bar{x} \in \bar{R}$ . Since $R$ is semiperfect, idempotents lift modulo $J$ : choose an idempotent $e \in R$ with $\bar{e} = \bar{x}$ . By $(21.18)$ , $e$ is a local idempotent, hence primitive, so $e \in E$ ; and $e R / e J ≅ \bar{e} \bar{R} ≅ V$ . Finally, $\bar{R}$ semisimple has only finitely many simple right modules up to isomorphism, which bounds both lists by the same finite number $r$ .

(2). Put $V_{i} = e_{i} R / e_{i} J$ . The module $P / P J$ is a finitely generated module over the semisimple ring $\bar{R}$ , hence $P / P J ≅ n_{1} V_{1} \oplus \dots \oplus n_{r} V_{r}$ with the $n_{i}$ uniquely determined (multiplicities of simple summands over a semisimple ring are invariants). Set $Q = n_{1} (e_{1} R) \oplus \dots \oplus n_{r} (e_{r} R)$ . Then $Q$ is finitely generated projective and $Q / Q J ≅ n_{1} V_{1} \oplus \dots \oplus n_{r} V_{r} ≅ P / P J$ , so $(19.27)$ gives $P ≅ Q$ . Uniqueness of the $n_{i}$ is inherited from uniqueness downstairs.

(3). By $(25.1)$ each ${End}_{R} (e_{i} R) ≅ e_{i} R e_{i}$ is local, so each $e_{i} R$ is strongly indecomposable. Thus $P$ is a finite direct sum of strongly indecomposable modules, and ${End}_{R} (P)$ is semiperfect by $(23.8)$ .

Corollary—Counting

Write $\bar{R} ≅ M_{n_{1}} (D_{1}) \times \dots \times M_{n_{r}} (D_{r})$ with $D_{i}$ division rings. Then $R$ has exactly $r$ principal indecomposables up to isomorphism, and in any decomposition $1 = e_{1} + \dots + e_{n}$ into orthogonal primitive idempotents the module $e_{i} R$ of type $i$ occurs exactly $n_{i}$ times; in particular $n = n_{1} + \dots + n_{r}$ .

Proof

Apply $(25.3) (2)$ to $P = R_{R}$ : the multiplicities are read off from $R / R J = \bar{R}$ , whose decomposition as a right module over itself is $n_{1} V_{1} \oplus \dots \oplus n_{r} V_{r}$ with $V_{i}$ the simple right $M_{n_{i}} (D_{i})$ -module. Since $R_{R} = e_{1} R \oplus \dots \oplus e_{n} R$ by $(23.6)$ and this decomposition is into indecomposables with local endomorphism rings, Krull–Schmidt–Azumaya makes the multiplicities unambiguous.

Remark—Krull–Schmidt at the right level of generality

Finite generation is not decoration. Both applications of $(19.27)$ above run through Nakayama's Lemma, applied to $P$ and then to a direct summand of it, and both need finite generation. Statements about arbitrary projective modules over semiperfect rings are genuinely harder and are not asserted here.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Reduce modulo the radical, then lift

Compute in the semisimple ring $\bar{R}$ , where everything is Wedderburn–Artin, then transport the answer up using $(19.27)$ for modules and idempotent lifting for elements.

Move 2

Nakayama to promote surjections

If $𝔄 + P J = P$ with $P$ finitely generated, then $𝔄 = P$ . This is what makes $P J$ small, hence what makes $P ↠ P / P J$ a projective cover.

Move 3

Turn module questions into corner-ring questions

Use ${End}_{R} (e R) ≅ e R e$ and ${Hom}_{R} (e R, M) ≅ M e$ . Indecomposability, local endomorphism rings and composition multiplicities all become statements about $e R e$ or about $M e$ .

Move 1 is the template for the entire chapter: semiperfectness is exactly the hypothesis that makes reduce-then-lift legitimate. Semilocal rings allow the reduction but not the lifting, and that is precisely where the theory of principal indecomposables stops.

Worked Example

Upper triangular matrices over a division ring

Let $k$ be a division ring and $R = T_{3} (k)$ the ring of upper triangular $3 \times 3$ matrices over $k$ , so $\dim_{k} R = 6$ . Being finite-dimensional, $R$ is right artinian, hence semiperfect. Its radical is the set of strictly upper triangular matrices:

J = rad R = (\begin{matrix} 0 & k & k \\ 0 & 0 & k \\ 0 & 0 & 0 \end{matrix}), J^{3} = 0, \bar{R} ≅ k \times k \times k .

(E.1)

Take $e_{i} = E_{i i}$ for $i = 1, 2, 3$ . Then $e_{i} R e_{i} = k E_{i i} ≅ k$ is a division ring, so each $e_{i}$ is a local idempotent, and $1 = e_{1} + e_{2} + e_{3}$ is an orthogonal decomposition. The module $P_{i} : = e_{i} R$ consists of the matrices of $R$ whose rows other than the $i$ -th vanish, so

\dim_{k} P_{1} = 3, \dim_{k} P_{2} = 2, \dim_{k} P_{3} = 1, R_{R} = P_{1} \oplus P_{2} \oplus P_{3} .

(E.2)

The simple tops are one-dimensional: $V_{i} = P_{i} / P_{i} J ≅ k$ , with $(a_{p q}) \in R$ acting by right multiplication by the diagonal entry $a_{i i}$ . Since $V_{1}, V_{2}, V_{3}$ are pairwise non-isomorphic, $P_{1}, P_{2}, P_{3}$ are pairwise non-isomorphic, and by $(25.3) (1)$ they form a complete list: $r = 3$ , and each multiplicity $n_{i}$ equals $1$ .

One checks that $e_{i} J ≅ e_{i + 1} R$ for $i < 3$ , which gives the unique composition series of each $P_{i}$ :

P_{1} ⊋ P_{1} J ⊋ P_{1} J^{2} ⊋ 0, factors V_{1}, V_{2}, V_{3}; P_{2} ⊋ P_{2} J ⊋ 0, factors V_{2}, V_{3} .

(25.11)

$P_{3} = V_{3}$ is already simple. Composition lengths are $3, 2, 1$ , matching the $k$ -dimensions because each $V_{i}$ is one-dimensional.

The same ring from the left

$P_{1} = e_{1} R$ is a two-sided ideal of $R$ . Viewed as a left module it is not indecomposable: $e_{1} R = k E_{11} \oplus k E_{12} \oplus k E_{13}$ , and left multiplication acts only on the first index, so all three summands are isomorphic to the simple left module $R e_{1} = k E_{11}$ . Indecomposability is a statement about one side only.

A contrasting case

For $R = M_{n} (k)$ with $k$ a local ring, $e = E_{11}$ is primitive and $e R$ is the space of matrices supported in the first row. Here $r = 1$ : there is a single principal indecomposable up to isomorphism, and $R_{R} ≅ n (e R)$ . The multiplicity $n_{1} = n$ is exactly the matrix size appearing in $\bar{R} ≅ M_{n} (k / rad k)$ .

Process and Workflow

Compute $J = rad R$ For a finite-dimensional algebra this is a linear-algebra computation; for a triangular or local ring it is usually visible by inspection.

Decompose $\bar{R} = R / J$ Wedderburn–Artin gives

\bar{R} ≅ \prod_{i = 1}^{r} M_{n_{i}} (D_{i})

; the number

r

is the number of principal indecomposables.

Lift a set of matrix idempotentsPick

{\bar{e}}_{i}

right irreducible in the

i

-th factor and lift to orthogonal idempotents

e_{i} \in R

. Semiperfectness guarantees this is possible.

Read off the modules

P_{i} = e_{i} R

are the principal indecomposables,

V_{i} = P_{i} / P_{i} J

the simple right modules, and

R_{R} ≅ n_{1} P_{1} \oplus \dots \oplus n_{r} P_{r}

You have a finitely generated projective right $R$ -module $P$ . How do you identify it?

Reduce mod $J$ Decompose the semisimple module

P / P J

; the multiplicity of

V_{i}

there is exactly the multiplicity

n_{i}

P_{i}

P

Test with idempotents

\dim {Hom}_{R} (e_{i} R, P) = \dim P e_{i}

over the relevant division ring — a rank computation rather than a decomposition.

$P$ not finitely generatedThe argument via Nakayama and

(19.27)

is unavailable. Do not assume unique decomposition; treat the infinite case separately.

Comparison and Classification

Principal indecomposables in familiar semiperfect rings
Ring $R$	$\bar{R} = R / rad R$	Number $r$ of PIMs	A typical PIM
Division ring $k$	$k$	1	$R$ itself, simple
Local ring $k$ (e.g. $k [[x]]$ )	$k / rad k$	1	$R$ itself, not simple unless $rad k = 0$
$M_{n} (k)$ , $k$ local	$M_{n} (k / rad k)$	1	a row space, occurring $n$ times in $R_{R}$
$T_{n} (k)$ , $k$ a division ring	$k \times \dots \times k$ ( $n$ factors)	$n$	$e_{i} R$ , of length $n - i + 1$
$\prod_{i = 1}^{t} M_{n_{i}} (D_{i})$ semisimple	itself	$t$	a minimal right ideal, i.e. a simple module
$ℤ / p^{m} ℤ$	$𝔽_{p}$	1	$R$ itself, of length $m$
$k G$ , $char k = p$ dividing $\| G \|$	semisimple, not $k G$	number of $p$ -regular classes over a splitting field	the projective cover of a simple $k G$ -module

Which properties of $e R$ hold under which hypotheses on $R$
	Any ring	Semilocal	Semiperfect	Right artinian
$e R$ projective and cyclic for $e = e^{2}$	yes	yes	yes	yes
$e$ primitive $\Rightarrow$ $e R e$ local	no	no	yes	yes
$e R / e J$ simple for $e$ primitive	no	no	yes	yes
$1$ a finite sum of orthogonal primitive idempotents	no	no	yes	yes
f.g. projectives decompose uniquely into PIMs	no	no	yes	yes
$e R$ has finite composition length	no	no	no	yes

Which properties of $e R$ hold under which hypotheses on $R$

Relationship Map

The three lists — primitive idempotents up to isomorphism, principal indecomposables up to isomorphism, simple right modules up to isomorphism — are the same list seen three ways.

e \in E

mod

≅

e R

mod

≅

e R / e J

mod

≅

simple

\bar{R}

-modules

$e R$ for $e \in E$ , $R$ semiperfect — principal indecomposable
- is
  - cyclic, generated by $e$
  - projective, a summand of $R_{R}$
  - strongly indecomposable: ${End}_{R} (e R) ≅ e R e$ is local
  - the projective cover of $e R / e J$
- has
  - a unique maximal submodule $e J$
  - simple top $e R / e J$
  - ${Hom}_{R} (e R, M) ≅ M e$ for every $M$
- need not be
  - of finite length (unless $R$ is right artinian)
  - indecomposable as a left module
  - isomorphic to $R e$ in any sense

Downstream, the same list indexes the rows and columns of the Cartan matrix, the summands of a basic idempotent, and the blocks of $R$ after grouping by linkage.

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

Projective indecomposables of $k G$

When $char k = p$ divides $| G |$ , the principal indecomposables of $k G$ are the projective covers of the simple $k G$ -modules; their composition factors are recorded in the Cartan matrix and underpin Brauer character theory.

Quiver algebras

Paths from a vertex

For a path algebra $k Q$ modulo an admissible ideal, the primitive idempotents are the vertices $e_{v}$ , and $e_{v} A$ has $k$ -basis the images of the paths starting at $v$ . Every basic finite-dimensional algebra over an algebraically closed field arises this way.

Computer algebra

Module decomposition routines

Systems that decompose modules over finite-dimensional algebras compute the radical, split the semisimple quotient, lift idempotents, and then express projectives in terms of the $e_{i} A$ — literally the four steps of the workflow above.

Homological algebra

Minimal projective resolutions

Because $e R$ is the projective cover of its top, syzygies over a semiperfect ring are computed by repeatedly covering the top of a module; minimal resolutions exist and are unique up to isomorphism.

The honest description is that principal indecomposables are infrastructure for representation theory: they are the objects an algorithm actually manipulates when it is asked about modules over an artinian algebra.

Computational Notes

Computational notes cover algorithms, cost and library behaviour rather than manufacturing process.

Let $A$ be a finite-dimensional algebra over a field $k$ with $\dim_{k} A = m$ , given by structure constants.

Computing $rad A$ costs $O (m^{3})$ field operations in characteristic $0$ (nullspace of the trace form); in characteristic $p$ the Friedl–Rónyai method is used and remains polynomial.
Splitting $A / rad A$ into simple factors requires factoring the centre and, over non-closed fields, identifying the division rings $D_{i}$ ; this is the expensive step in practice.
Idempotent lifting modulo a nilpotent radical is Newton-like: from $e$ with $e^{2} - e \in J$ , the element $3 e^{2} - 2 e^{3}$ improves the defect from $J^{t}$ to $J^{2 t}$ , so $O (\log \dim A)$ rounds suffice.
Once the $e_{i}$ are known, $\dim_{k} {Hom}_{A} (e_{i} A, M) = \dim_{k} M e_{i}$ turns every multiplicity question into a rank computation on the matrix of right multiplication by $e_{i}$ .

Failure Modes and Common Mistakes

Do not conflate indecomposable with strongly indecomposable: the second means the endomorphism ring is local and is what Krull–Schmidt actually needs.
Do not assume $e R$ has a composition series. It does when $R$ is right artinian; over a general semiperfect ring it need only have a simple top.
Do not read a right-module statement as a left-module statement. A right principal indecomposable can decompose badly as a left module, as $e_{1} T_{n} (k)$ shows.
Do not confuse the multiplicity $n_{i}$ of $e_{i} R$ inside $R_{R}$ with the composition multiplicity of $V_{j}$ inside $e_{i} R$ ; the first is a matrix size in $\bar{R}$ , the second is a Cartan invariant.

Best Practices

Fix a side at the start of any computation and state it; every module in sight is then unambiguous.
Record the pair $(P_{i}, V_{i})$ together — the projective and its top — rather than either alone; almost every later statement uses both.
Check $\sum_{i} n_{i} \dim V_{i} = \dim \bar{R}$ as a cheap consistency test for a claimed list of principal indecomposables.
When a ring is presented by matrices, look for the diagonal matrix units first: they are usually already a complete orthogonal set of local idempotents.

Quick Reference

Setting

R

semiperfect,

J = rad R

\bar{R} = R / J

semisimple,

E

the primitive idempotents

PIM

e R

for

e \in E

; cyclic, projective, strongly indecomposable

Top

e R / e J

simple;

e J = rad (e R)

is the unique maximal submodule

Endomorphisms

{End}_{R} (e R) ≅ e R e

, a local ring

Hom formula

{Hom}_{R} (e R, M) ≅ M e

Classification

e R \mapsto e R / e J

is a bijection onto the simple right

R

-modules

Projectives

P

f.g. projective

\Rightarrow

P ≅ ⨁_{i} n_{i} (e_{i} R)

, the

n_{i}

unique

Endomorphism ring

{End}_{R} (P)

is again semiperfect

Statement locator
Fact	Statement	Reference
Primitive is local	$e$ primitive in semiperfect $R$ $\Rightarrow$ $e R e$ local	(25.1), (23.5)
Simple top	$e R / e J ≅ \bar{e} \bar{R}$ simple; $e J$ unique maximal	(25.2), (21.18)
Bijection	PIMs $\leftrightarrow$ simple right modules	(25.3)(1)
Unique decomposition	f.g. projective $= ⨁ n_{i} (e_{i} R)$	(25.3)(2)
Semiperfect endomorphisms	${End}_{R} (P)$ semiperfect	(25.3)(3), (23.8)
Lifting lemma	$P ≅ Q$ iff $P / P J ≅ Q / Q J$ for f.g. projectives	(19.27)
Idempotent decomposition	$1 = e_{1} + \dots + e_{n}$ , all $e_{i}$ local	(23.6)

Frequently Asked Questions

Why are they called principal indecomposables?

Because they are principal right ideals — $e R$ is generated by the single element $e$ — that happen to be indecomposable as modules. The name predates the general theory and comes from the classical study of right artinian rings, where these modules were first isolated.

Is every indecomposable projective right module a principal indecomposable?

Every finitely generated one is, over a semiperfect ring: it is a summand of a free module of finite rank, and $(25.3) (2)$ decomposes it into modules $e_{i} R$ ; indecomposability forces exactly one summand. Without finite generation this argument is unavailable and the statement should not be assumed.

How do I tell whether

e R ≅ f R

without decomposing anything?

Use the idempotent criterion $(21.20)$ : $e R ≅ f R$ as right modules if and only if there exist $a \in e R f$ and $b \in f R e$ with $a b = e$ and $b a = f$ . Equivalently, reduce modulo the radical and compare $\bar{e} \bar{R}$ with $\bar{f} \bar{R}$ inside the semisimple ring $\bar{R}$ .

Does

e R

determine

R e

The idempotent $e$ determines both, and $e R ≅ f R$ holds if and only if $R e ≅ R f$ , since both are equivalent to $e$ and $f$ being isomorphic idempotents. But the modules themselves can look completely different: over $T_{n} (k)$ the right module $e_{1} R$ has length $n$ while the left module $R e_{1}$ is simple.

What replaces this theory when

R

is only semilocal?

Very little. Semilocal rings give a semisimple quotient $\bar{R}$ , so simple modules are still classified, but without idempotent lifting there is no way back: primitive idempotents of $R$ need not be local, and there may be no finite orthogonal decomposition of $1$ into primitive idempotents at all.

How does the composition length of

e R

relate to the Cartan matrix?

When $R$ is right artinian, the length of $e_{i} R$ is the sum of the $i$ -th row of the Cartan matrix, since the $(i, j)$ entry counts the composition factors of $e_{i} R$ isomorphic to $V_{j}$ . The diagonal entries are always at least $1$ because $V_{i}$ appears as the top of $e_{i} R$ .

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §25 (pp. 370–373).
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §27.
H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (1960), 466–488.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley-Interscience, 1981, Chapter 6.
L. H. Rowen, Ring Theory, Volume I, Academic Press, 1988, Chapter 2.

AI Suggested Questions

Show that a local idempotent is primitive in any ring, and give an example of a primitive idempotent that is not local.
Work out the principal indecomposable modules of $k Q$ for $Q$ the quiver with two vertices and one arrow, and compare with $T_{2} (k)$ .
Prove that $e R$ is the projective cover of $e R / e J$ directly from the definition of a projective cover.
Which of the conclusions of $(25.3)$ survive if $R$ is only assumed semilocal?
Describe the principal indecomposable modules of $ℤ_{p} [C_{p}]$ for the cyclic group of order $p$ over the $p$ -adic integers.
How does the list of principal indecomposables change under a field extension $k \subseteq K$ for a finite-dimensional $k$ -algebra?
Explain the role of Azumaya's version of Krull–Schmidt in making the multiplicities $n_{i}$ well defined.

Principal Indecomposables

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

From idempotents to modules and back

Why the top is simple

Projective covers

Key Results

Proof Techniques and Method

Reduce modulo the radical, then lift

Nakayama to promote surjections

Turn module questions into corner-ring questions

Worked Example

Upper triangular matrices over a division ring

The same ring from the left

A contrasting case

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Projective indecomposables of $k G$

Paths from a vertex

Module decomposition routines

Minimal projective resolutions

Computational Notes

Failure Modes and Common Mistakes

Best Practices

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

From idempotents to modules and back

Why the top is simple

Projective covers

06Key Results

07Proof Techniques and Method

Reduce modulo the radical, then lift

Nakayama to promote surjections

Turn module questions into corner-ring questions

08Worked Example

Upper triangular matrices over a division ring

The same ring from the left

A contrasting case

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Projective indecomposables of kG

Paths from a vertex

Module decomposition routines

Minimal projective resolutions

13Computational Notes

14Failure Modes and Common Mistakes

15Best Practices

16Quick Reference

17Frequently Asked Questions

18Related KEVOS Topics

19References

20AI 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

Projective indecomposables of $k G$

Computational Notes

Failure Modes and Common Mistakes

Best Practices

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions