Semiperfect Endomorphism Rings and Module Decompositions

Q: How does (23.9) improve on (23.2)?

(23.2) handles M_n(k) for k local, proved by hand through diagonalisation over the residue division ring. (23.9) removes the locality hypothesis: k need only be semiperfect. The proof is shorter because (23.8) has already done the work.

Q: Is semiperfectness a Morita invariant?

Yes. k is semiperfect iff M_m(k) is, and more generally the property is characterised by the existence of finite decompositions with local endomorphism rings in the module category, which is preserved by any category equivalence.

Q: If End(M) is semiperfect, what do I learn about k?

Nothing directly. Take k = Z, which is not semiperfect, and M any finite abelian group: End(M) is a finite ring and hence semiperfect. The theorem constrains M, not the base.

Executive Summary

Semiperfect rings were defined by a property of $R / rad R$ and then characterised by a decomposition of $1$ . This page gives the third face of the same notion, and the one that explains where semiperfect rings come from: they are exactly the endomorphism rings of modules that decompose into finitely many pieces with local endomorphism rings.

The correspondence is mechanical in both directions. A decomposition $M = M_{1} \oplus \dots \oplus M_{n}$ gives orthogonal projections $e_{i} \in R = End (M_{k})$ with $e_{i} R e_{i} ≅ End (M_{i})$ ; a decomposition of $1 \in R$ into orthogonal local idempotents gives back the summands $M_{i} = e_{i} (M)$ . Locality of the corner and strong indecomposability of the summand are literally the same statement.

(23.8)The equivalence

$e_{i} R e_{i} ≅ End (M_{i})$ The dictionary

FiniteNumber of summands

(23.9)

M_{n} (k)

semiperfect

Overview

Fix a ring $k$ and a right $k$ -module $M$ , and write $R = End (M_{k})$ , with endomorphisms composing on the left so that $M$ is an $(R, k)$ -bimodule. Direct decompositions of $M$ correspond to complete orthogonal families of idempotents of $R$ ; this correspondence is a triviality. What is not trivial is that under it, local idempotents match strongly indecomposable summands.

M = M_{1} \oplus \dots \oplus M_{n} ⟷ 1 = e_{1} + \dots + e_{n} in R, M_{i} = e_{i} (M), e_{i} R e_{i} ≅ End (M_{i})

(23.8a)

The dictionary underlying the whole page.

Combining this with the idempotent characterisation of semiperfectness gives $(23.8)$ immediately. The theorem is therefore not deep in its proof; it is valuable because of what it lets one import. Module-theoretic hypotheses — finite length, finite direct sums of indecomposables, uniqueness of decomposition — become ring-theoretic ones, and results proved for semiperfect rings become decomposition theorems for modules.

The construction is also general enough to produce every semiperfect ring: given any semiperfect $R$ , take $k = R$ and $M = R_{R}$ . Then $M = e_{1} R \oplus \dots \oplus e_{n} R$ with each $e_{i} R$ strongly indecomposable, and $R ≅ End (R_{R})$ . So $(23.8)$ is a complete description, not merely a source of examples.

Learning Objectives

Verify that a decomposition of $M$ produces orthogonal idempotents of $End (M_{k})$ summing to $1$ .
Prove $e_{i} R e_{i} ≅ End (M_{i})$ and describe $e_{i} R e_{j}$ for $i \neq j$ .
Prove both directions of $(23.8)$ using the idempotent characterisation $(23.6)$ .
Deduce $(23.9)$ : $M_{n} (k)$ is semiperfect whenever $k$ is.
Explain why finite length forces $End (M)$ to be semiperfect.
Give a module that is indecomposable but not strongly indecomposable and say what goes wrong.

Definitions

Definition—Strongly indecomposable module

A nonzero module $N$ is strongly indecomposable if $End (N)$ is a local ring. Since a local ring has no idempotents other than $0$ and $1$ , and idempotent endomorphisms are exactly the projections onto direct summands, a strongly indecomposable module is indecomposable. The converse fails.

$End (M_{k})$: The ring of $k$ -endomorphisms of the right $k$ -module $M$ . Multiplication is composition; the identity is ${id}_{M}$ .
$e_{i}$: The projection attached to a fixed decomposition: $e_{i}$ restricts to the identity on $M_{i}$ and kills every $M_{j}$ with $j \neq i$ .
$e_{i} R e_{j}$: The set of $f \in R$ with $f (M_{j}) \subseteq M_{i}$ and $f (M_{l}) = 0$ for $l \neq j$ ; naturally isomorphic to $Hom (M_{j}, M_{i})$ .
Finite length: A module with a composition series. Such modules are finite direct sums of strongly indecomposable modules by Fitting's Lemma.

No chain condition is imposed on $k$ or on $M$ anywhere in this section; that is precisely the point of stating the theorem for arbitrary modules.

Core Concepts

Idempotents are decompositions

If $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ orthogonal idempotents of $R = End (M_{k})$ , then every $x \in M$ satisfies $x = e_{1} (x) + \dots + e_{n} (x)$ , and $e_{i} (M) \cap \sum_{j \neq i} e_{j} (M) = 0$ because applying $e_{i}$ to an element of the right-hand sum gives $0$ while fixing an element of $e_{i} (M)$ . Hence $M = e_{1} (M) \oplus \dots \oplus e_{n} (M)$ . Conversely a direct decomposition defines its projections. The correspondence is bijective and inverse to itself.

The Peirce description of the corners

Fix $M = M_{1} \oplus \dots \oplus M_{n}$ with projections $e_{i}$ . Then

e_{i} R e_{j} = {f \in R : f (M_{j}) \subseteq M_{i} and f (M_{l}) = 0 for all l \neq j} ≅ {Hom}_{k} (M_{j}, M_{i}),

(23.8b)

The Peirce decomposition $R = ⨁_{i, j} e_{i} R e_{j}$ realises $R$ as a matrix ring of Hom-groups.

In particular $e_{i} R e_{i} ≅ End (M_{i})$ as rings, the isomorphism being restriction of $f$ to $M_{i}$ . This single identification carries the whole theorem.

Why strongly indecomposable and not merely indecomposable

Indecomposability of $M_{i}$ says only that $End (M_{i})$ has no idempotents besides $0$ and $1$ — that is, $e_{i}$ is primitive. Semiperfectness needs the corner to be local, which is strictly stronger. The gap is real: $ℤ$ is an indecomposable $ℤ$ -module with $End (ℤ) = ℤ$ , a ring that is not local, and $End (ℤ \oplus ℤ) = M_{2} (ℤ)$ is not semiperfect.

Key Results

Theorem(23.8)Semiperfect endomorphism rings

Let $k$ be a ring and $M$ a right $k$ -module, and set $R = End (M_{k})$ . Then $M$ is a finite direct sum of strongly indecomposable $k$ -modules if and only if $R$ is a semiperfect ring.

Proof

**( $\Rightarrow$ )** Write $M = M_{1} \oplus \dots \oplus M_{n}$ with each $End (M_{i})$ local, and let $e_{i} \in R$ be the associated projections. They are mutually orthogonal idempotents with $e_{1} + \dots + e_{n} = {id}_{M} = 1$ . By $(23.8 b)$ , $e_{i} R e_{i} ≅ End (M_{i})$ is local, so each $e_{i}$ is a local idempotent. The criterion $(23.6)$ now says $R$ is semiperfect.

**( $\Leftarrow$ )** Let $R$ be semiperfect. By $(23.6)$ there are mutually orthogonal local idempotents with $1 = e_{1} + \dots + e_{n}$ . Put $M_{i} = e_{i} (M)$ . As shown above, $M = M_{1} \oplus \dots \oplus M_{n}$ , and each $M_{i}$ is nonzero because $e_{i} \neq 0$ . Restriction gives $End (M_{i}) ≅ e_{i} R e_{i}$ , which is local by hypothesis, so every $M_{i}$ is strongly indecomposable.

Note that no finiteness assumption on $M$ or on $k$ enters; the finiteness is entirely in the number of summands, which is exactly what $(23.6)$ supplies.

Remark—Every semiperfect ring arises this way

Let $R$ be semiperfect with $1 = e_{1} + \dots + e_{n}$ as in $(23.6)$ . Take $k = R$ and $M = R_{R}$ , the right regular module. Then $M = e_{1} R \oplus \dots \oplus e_{n} R$ , each $End (e_{i} R) ≅ e_{i} R e_{i}$ is local, and $R ≅ End (R_{R})$ . So $(23.8)$ characterises the class rather than merely describing part of it.

Corollary(23.9)Matrix rings over semiperfect rings

If $k$ is a semiperfect ring, then $M_{m} (k)$ is semiperfect for every $m \geq 1$ .

Proof

Identify $M_{m} (k)$ with $End ((k^{m})_{k})$ , where $k^{m}$ is the free right $k$ -module of rank $m$ . Since $k$ is semiperfect, $(23.6)$ gives $1 = e_{1} + \dots + e_{n}$ with local idempotents, whence $k_{k} = e_{1} k \oplus \dots \oplus e_{n} k$ with $End (e_{i} k) ≅ e_{i} k e_{i}$ local. Thus $k_{k}$ is a finite direct sum of strongly indecomposable modules, and so is $(k^{m})_{k}$ — a direct sum of $m$ copies, that is $m n$ strongly indecomposable summands. Apply $(23.8)$ to $M = (k^{m})_{k}$ .

This generalises $(23.2)$ , which is the case of $k$ local ( $n = 1$ ).

Corollary—Morita invariance

$k$ is semiperfect if and only if $M_{m} (k)$ is semiperfect. One direction is $(23.9)$ ; for the other, note $k ≅ e M_{m} (k) e$ for the idempotent $e = E_{11}$ , and a corner $e R e$ of a semiperfect ring $R$ at a nonzero idempotent $e$ is again semiperfect. More generally semiperfectness is preserved by Morita equivalence, since it is characterised by a property of the module category — the existence of finite decompositions with local endomorphism rings.

Corollary—Finite length and finitely generated projectives

(a) If $M$ has finite length as a $k$ -module then $End (M_{k})$ is semiperfect: $M$ decomposes into finitely many indecomposables, and Fitting's Lemma makes each of their endomorphism rings local.

(b) If $R$ is semiperfect and $P$ is a finitely generated projective right $R$ -module, then $End (P)$ is semiperfect. Indeed $P$ is a direct summand of some $R^{m} = ⨁_{i} (e_{i} R)^{m}$ , a finite direct sum of modules with local endomorphism rings, so by the Krull–Schmidt–Azumaya theorem $P$ is itself a finite direct sum of copies of the $e_{i} R$ ; now apply $(23.8)$ .

Proof Techniques and Method

How these proofs work, and which move is reusable.

Move 1

Translate, do not compute

Both directions of $(23.8)$ are dictionary lookups: projections become idempotents, summands become corners. Once $(23.6)$ is available there is nothing left to prove.

Move 2

Realise the ring as an endomorphism ring

To prove a ring semiperfect, exhibit it as $End (M_{k})$ for a module you can decompose. $(23.9)$ is this move applied to $M_{m} (k) = End ((k^{m})_{k})$ .

Move 3

Pass to a summand via Azumaya

Summands of finite direct sums of local-endomorphism modules are again such sums. This is what lets results transfer from $R$ to $End (P)$ for $P$ finitely generated projective.

Move 2 is the practical one. Many rings that do not look semiperfect are endomorphism rings in disguise: triangular matrix rings, incidence algebras of finite posets over a field, and endomorphism rings of finite-length modules over any base.

The recurring error is to check indecomposability instead of strong indecomposability. Endomorphism rings, not summand counts, are what the theorem is about.

Worked Example

A finite abelian group

Take $k = ℤ$ and $M = ℤ / 4 \oplus ℤ / 2$ . Each summand is strongly indecomposable: $End (ℤ / p^{a}) ≅ ℤ / p^{a}$ , a local ring with radical $(p)$ . By $(23.8)$ , $R = End (M)$ is semiperfect. Concretely, using $(23.8 b)$ ,

R ≅ (\begin{matrix} End (ℤ / 4) & Hom (ℤ / 2, ℤ / 4) \\ Hom (ℤ / 4, ℤ / 2) & End (ℤ / 2) \end{matrix}) ≅ (\begin{matrix} ℤ / 4 & ℤ / 2 \\ ℤ / 2 & ℤ / 2 \end{matrix}),

(E.1)

A ring of order $4 \cdot 2 \cdot 2 \cdot 2 = 32$ , written in Peirce form.

The off-diagonal parts multiply into the radical: if $f : ℤ / 2 \to ℤ / 4$ and $g : ℤ / 4 \to ℤ / 2$ , then $f (1) \in {0, 2}$ , so $g f = 0$ and $(f g)^{2} = 0$ . Hence

rad R = (\begin{matrix} 2 ℤ / 4 & ℤ / 2 \\ ℤ / 2 & 0 \end{matrix}), | rad R | = 8, R / rad R ≅ 𝔽_{2} \times 𝔽_{2} .

(E.2)

$32 / 8 = 4$ : the quotient is semisimple, as $(23.8)$ predicts.

The decomposition of $1$ is $e_{1} + e_{2}$ , the two projections, with corners $ℤ / 4$ and $ℤ / 2$ — both local, both non-isomorphic, so $R$ has exactly two simple modules.

The same construction failing

Now take $M = ℤ \oplus ℤ$ over $k = ℤ$ . The summand $ℤ$ is indecomposable but $End (ℤ) = ℤ$ is not local, so $M$ is not a finite direct sum of strongly indecomposable modules — and indeed no other decomposition helps, since every decomposition of $ℤ^{2}$ into indecomposables has summands isomorphic to $ℤ$ . Consistently, $End (M) = M_{2} (ℤ)$ is not semiperfect: $rad M_{2} (ℤ) = M_{2} (0) = 0$ and $M_{2} (ℤ)$ is not semisimple.

An infinite decomposition

Let $k$ be a field and $M = k^{(ℕ)}$ , a countable direct sum of copies of $k$ . Each summand is strongly indecomposable ( $End (k) = k$ ), but there are infinitely many of them. $End (M)$ is the ring of column-finite $ℕ \times ℕ$ matrices over $k$ , which contains an infinite orthogonal family of nonzero idempotents and is therefore not semiperfect. Finiteness of the number of summands is essential in $(23.8)$ .

Frameworks and Models

It helps to keep three parallel columns in mind: a property of the module, the matching property of the idempotent, and the matching property of the ring.

The module–idempotent–ring dictionary
Module side	Idempotent side	Ring side
Direct summand $M_{i}$	Idempotent $e_{i}$	Corner $e_{i} R e_{i}$
$M_{i}$ indecomposable	$e_{i}$ primitive	$e_{i} R e_{i}$ has only trivial idempotents
$M_{i}$ strongly indecomposable	$e_{i}$ local	$e_{i} R e_{i}$ local
Finite decomposition of $M$	$1 = e_{1} + \dots + e_{n}$	complete orthogonal family
Finite decomposition into strongly indecomposables	orthogonal local idempotents summing to $1$	$R$ semiperfect
$Hom (M_{j}, M_{i})$	$e_{i} R e_{j}$	Peirce component

Modules $M$ with $End (M)$ semiperfect
- finite length modules
  - any f.g. module over a left artinian ring
  - any finite abelian group as a $ℤ$ -module
- finitely generated projectives over a semiperfect ring
  - principal indecomposables $e_{i} R$
  - finite direct sums of them
- modules with local endomorphism ring
  - $ℚ$ over $ℤ$
  - $ℤ / p^{n}$ over $ℤ$
  - any uniserial module of finite length

Comparison and Classification

Which modules have semiperfect endomorphism rings
	Decomposes finitely	Summands strongly indecomposable	$End (M)$ semiperfect
$M$ of finite length over any $k$	yes	yes	yes
$ℤ / 4 \oplus ℤ / 2$ over $ℤ$	yes	yes	yes
$k^{m}$ over a semiperfect $k$	yes	yes	yes
$ℚ$ over $ℤ$	yes	yes	yes
$ℤ \oplus ℤ$ over $ℤ$	yes	no	no
$k^{(ℕ)}$ over a field $k$	no	yes	no
$ℚ \oplus ℤ$ over $ℤ$	yes	partial	no

Which modules have semiperfect endomorphism rings

Endomorphism rings of familiar modules
$k$	$M$	$End (M_{k})$	Semiperfect?
field $k$	$k^{n}$	$M_{n} (k)$	yes
$ℤ_{p}$	$ℤ_{p}^{n}$	$M_{n} (ℤ_{p})$	yes
$ℤ$	$ℤ / p^{a_{1}} \oplus \dots \oplus ℤ / p^{a_{r}}$	finite ring, Peirce form	yes
$ℤ$	$ℤ^{n}$	$M_{n} (ℤ)$	no
field $k$	$k^{(ℕ)}$	column-finite matrices	no
$k [x]$	$k [x]$	$k [x]$	no

Relationship Map

End (N)

local

N

strongly indecomposable

N

indecomposable

The first arrow is a definition and the second is strict; equality of the last two notions is exactly what Fitting's Lemma buys under a finite-length hypothesis — see Strongly Indecomposable Modules and Local Endomorphism Rings.

Decompose $M$ Find

M = M_{1} \oplus \dots \oplus M_{n}

with

n

finite. Any decomposition will do at this stage.

Test each $End (M_{i})$ Local, not merely idempotent-free. Over a finite-length module this is automatic; otherwise it must be checked.

Conclude semiperfectness

(23.8)

upgrades the decomposition to a ring-theoretic statement about

End (M)

Import the consequencesKrull–Schmidt uniqueness for the decomposition, projective covers for

End (M)

-modules, and a finite list of simples.

Uniqueness travels with the theorem: by Krull–Schmidt–Azumaya, a decomposition into modules with local endomorphism rings is unique up to isomorphism and reordering, which matches the uniqueness of the idempotent decomposition in $(23.7) (1)$ under the dictionary.

Applications and Industry Use

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

Integral representation theory

Lattices over orders

For a $ℤ_{p}$ -order $Λ$ and a $Λ$ -lattice $L$ , $End (L)$ is semiperfect, which is what makes the Krull–Schmidt theorem valid for $p$ -adic lattices — and false, notoriously, for lattices over $ℤ$ .

Quiver and poset algebras

Incidence algebras

The incidence algebra of a finite poset over a field is the endomorphism ring of a finite direct sum of local pieces; $(23.8)$ explains directly why such algebras are semiperfect and where their principal indecomposables come from.

Computer algebra

Module decomposition algorithms

Meataxe-style algorithms decompose a module by finding idempotents in its endomorphism ring. $(23.8)$ is the guarantee that the search terminates with local corners when the module has finite length.

Homological algebra

Minimal resolutions of modules

Resolving a module by projectives with local endomorphism rings gives minimality; the ambient hypothesis needed is exactly semiperfectness of the relevant endomorphism ring.

Honest summary: $(23.8)$ is a translation device. Its value is that decomposition questions about modules and structure questions about rings become the same question, so a technique developed on one side is immediately available on the other.

Design Considerations

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

Which object carries the hypothesis? If your data is a module, check strong indecomposability of summands; if it is a ring, check the decomposition of $1$ . $(23.8)$ says you may choose whichever is cheaper.
Side conventions. Writing endomorphisms on the left of a right module makes $M$ an $(R, k)$ -bimodule and keeps $e_{i} R e_{i} ≅ End (M_{i})$ a ring isomorphism rather than an anti-isomorphism. Mixing conventions silently introduces an opposite ring.
How much finiteness to assume. Finite length is far stronger than what $(23.8)$ needs. Assuming it discards genuinely infinite examples such as $ℚ$ over $ℤ$ , which is strongly indecomposable without being finite length.
Model the summands, not the whole. The Peirce form $(23.8 b)$ reduces computations in $End (M)$ to $Hom$ groups between summands, which is usually where the concrete linear algebra lives.

Computational Notes

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

Deciding indecomposability of a finite-dimensional module $M$ over a field reduces to finding a nontrivial idempotent in $End (M)$ , which is computed as the nullspace of a linear system of size $(\dim M)^{2}$ and then decomposed via its radical.
The Meataxe and its descendants split $M$ by exhibiting an endomorphism whose minimal polynomial factors; over a finite field this succeeds with high probability after a constant expected number of random choices.
Once a decomposition is found, $(23.8 b)$ gives $End (M)$ in Peirce block form, which is the compact representation used by computer algebra systems: store $Hom (M_{j}, M_{i})$ blocks rather than one large matrix algebra.
For modules over $ℤ$ or over an order, decomposition is genuinely harder — Krull–Schmidt can fail over non-complete bases, so a computed decomposition is not canonical and must be reported as one of possibly many.

Failure Modes and Common Mistakes

$(23.8)$ says nothing about $k$ itself. $End (M_{k})$ can be semiperfect over a wildly non-semiperfect $k$ — take $k = ℤ$ and $M$ any finite abelian group.
A module can have several decompositions into strongly indecomposables; they agree only up to isomorphism and permutation, by Krull–Schmidt–Azumaya, not on the nose.
Semiperfectness of $End (M)$ does not make $M$ finitely generated, noetherian or artinian. $ℚ$ over $ℤ$ is none of these and has $End (ℚ) = ℚ$ , a field.

Quick Reference

$(23.8)$

End (M_{k})

semiperfect

iff M

is a finite direct sum of strongly indecomposables

Dictionary

M_{i} = e_{i} (M)

e_{i} R e_{i} ≅ End (M_{i})

Peirce

e_{i} R e_{j} ≅ Hom (M_{j}, M_{i})

$(23.9)$

k

semiperfect

\Rightarrow M_{m} (k)

semiperfect

Converse

M_{m} (k)

semiperfect

\Rightarrow k ≅ E_{11} M_{m} (k) E_{11}

semiperfect

Automatic case

M

of finite length

\Rightarrow End (M)

semiperfect

Failure modesinfinitely many summands; summands merely indecomposable

UniquenessKrull–Schmidt–Azumaya, up to isomorphism and order

Results of this page
Reference	Statement	Hypotheses
(23.8)	$M$ a finite sum of strongly indecomposables $iff End (M_{k})$ semiperfect	$M$ any right $k$ -module; no chain conditions
(23.8b)	Peirce description of $e_{i} R e_{j}$	a fixed finite decomposition of $M$
(23.9)	$M_{m} (k)$ semiperfect	$k$ semiperfect, $m \geq 1$
—	$End (P)$ semiperfect	$R$ semiperfect, $P$ finitely generated projective
—	$End (M)$ semiperfect	$M$ of finite length over any ring

Frequently Asked Questions

Why is strongly indecomposable the right hypothesis rather than indecomposable?

Because the ring side needs a local corner, and a corner with only trivial idempotents need not be local. $ℤ$ is indecomposable over itself with $End (ℤ) = ℤ$ non-local, and $M_{2} (ℤ)$ is not semiperfect. Under a finite-length hypothesis the distinction evaporates by Fitting's Lemma.

Does

(23.8)

require

M

to be finitely generated?

No. It requires only that the number of summands be finite. $ℚ$ over $ℤ$ is not finitely generated, is strongly indecomposable, and has $End (ℚ) = ℚ$ , a field — semiperfect for the trivial reason that it is local.

How does

(23.9)

improve on

(23.2)

$(23.2)$ handles $M_{n} (k)$ for $k$ local, proved by hand through diagonalisation over the residue division ring. $(23.9)$ removes the locality hypothesis: $k$ need only be semiperfect. The proof is shorter because $(23.8)$ has already done the work.

Is semiperfectness a Morita invariant?

Yes. $k$ is semiperfect iff $M_{m} (k)$ is, and more generally the property is characterised by the existence of finite decompositions with local endomorphism rings in the module category, which is preserved by any category equivalence.

What does the theorem give me about uniqueness of decompositions?

Via Krull–Schmidt–Azumaya, a decomposition into modules with local endomorphism rings is unique up to isomorphism of summands and reordering. Under the dictionary this is exactly the statement that the idempotent decomposition of $1$ is unique up to conjugation and permutation.

End (M)

is semiperfect, what do I learn about

k

Nothing directly. Take $k = ℤ$ , which is not semiperfect, and $M$ any finite abelian group: $End (M)$ is a finite ring and hence semiperfect. The theorem constrains $M$ , not the base.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §23, results (23.8)–(23.9); Krull–Schmidt material in §19.
G. Azumaya, “Corrections and supplementaries to my paper concerning Krull–Remak–Schmidt's theorem”, Nagoya Mathematical Journal 1 (1950), 117–124.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, chapters on decompositions and semiperfect rings.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley, 1981, chapters on lattices over orders and the Krull–Schmidt theorem.
H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (1960), 466–488.

AI Suggested Questions

Prove that a direct summand of a finite direct sum of modules with local endomorphism rings is again such a sum.
Give an indecomposable module of infinite length whose endomorphism ring is local but not noetherian.
Why does Krull–Schmidt fail for lattices over $ℤ$ but hold over $ℤ_{p}$ ?
Compute $End (M)$ and its radical for $M = ℤ / p^{3} \oplus ℤ / p$ and identify the simple modules.
For which finite posets is the incidence algebra over a field a basic semiperfect ring?
What replaces $(23.8)$ when the decomposition of $M$ is infinite — how is the semiregular case stated?
How do Meataxe-style algorithms locate idempotents in $End (M)$ , and what is the failure probability?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Idempotents are decompositions

The Peirce description of the corners

Why strongly indecomposable and not merely indecomposable

06Key Results

07Proof Techniques and Method

Translate, do not compute

Realise the ring as an endomorphism ring

Pass to a summand via Azumaya

08Worked Example

A finite abelian group

The same construction failing

An infinite decomposition

09Frameworks and Models

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Lattices over orders

Incidence algebras

Module decomposition algorithms

Minimal resolutions of modules

13Design Considerations

14Computational Notes

15Failure Modes and Common Mistakes

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

Frameworks and Models

Comparison and Classification

Relationship Map

Applications and Industry Use

Design Considerations

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions