Idempotents and Direct Decompositions of Modules

Q: Why is Hom_R(eR, e'R) equal to e'Re and not eRe'?

Because the isomorphism is evaluation at the generator e of the source, and the value lies in the target e'R. So the value is an element m with m = e'm and m = me, that is, m ∈ e'Re. The mnemonic is that composition of maps must match multiplication of these sets, and (e''Re')(e'Re) ⊆ e''Re only typechecks with targets on the left.

Q: How do I recognise that two idempotents give isomorphic modules?

eR ≅ e'R as right R-modules if and only if there exist a ∈ eRe' and b ∈ e'Re with ab = e and ba = e'; equivalently, if and only if there exist a, b ∈ R with ab = e and ba = e'. This is Lam's (21.20), and it is proved by feeding a mutually inverse pair of homomorphisms through (21.6).

Q: Why insist on *strongly* indecomposable for local idempotents?

Because locality of End, not mere indecomposability, is what powers the Krull–Schmidt theorem and the theory of principal indecomposable modules. Over rings without finiteness conditions the two notions genuinely differ, and theorems that assume one do not survive substitution of the other.

Q: Does (21.6) need M to be projective, finitely generated, or anything else?

No. M is an arbitrary right R-module and e an arbitrary idempotent. The proof uses only e^2 = e and R-linearity. This generality is what makes the isomorphism a useful computational tool rather than a structural theorem.

Executive Summary

Direct sum decompositions and idempotents are the same thing viewed twice. A splitting $M = M_{1} \oplus M_{2}$ is exactly a projection $π \in End (M)$ with $π^{2} = π$ , and a splitting into $n$ pieces is exactly a complete orthogonal family of $n$ idempotents in $End (M)$ . Applying this to $M = R$ itself converts module theory into ring arithmetic.

The computational engine is a single natural isomorphism: ${Hom}_{R} (e R, M) ≅ M e$ . Its specialisation ${End}_{R} (e R) ≅ e R e$ says that the endomorphism ring of the projective module $e R$ is the corner ring at $e$ — so questions about how $e R$ decomposes become questions about idempotents inside $e R e$ .

$M e$

{Hom}_{R} (e R, M)

$e R e$

{End}_{R} (e R)

4Equivalent forms of primitivity

Local ⟹ primitiveStrictly one way

Overview

Fix a ring $R$ with identity and work with right $R$ -modules; homomorphisms are written on the left of their arguments. If $e$ is an idempotent then $R = e R \oplus (1 - e) R$ , so $e R$ is a direct summand of the free module $R$ and hence projective. Every projective module produced by an idempotent looks like this, and the point of the section is that its homological invariants are computed by multiplication in $R$ rather than by anything homological.

λ : {Hom}_{R} (e R, M) \tilde{⟶} M e, λ (θ) = θ (e) .

(21.6)

Natural in $M$ ; an isomorphism of additive groups, of $e R e$ -modules on one side and of $End (M)$ -modules on the other.

From this one gets ${Hom}_{R} (e R, e' R) ≅ e' R e$ and, taking $e' = e$ , the ring isomorphism ${End}_{R} (e R) ≅ e R e$ . Indecomposability of $e R$ then becomes the absence of nontrivial idempotents in $e R e$ , and the stronger condition that ${End}_{R} (e R)$ be local becomes the statement that $e R e$ is local. These two conditions name the two most important classes of idempotents: primitive and local.

Learning Objectives

Prove $(21.6)$ : evaluation at $e$ is an isomorphism ${Hom}_{R} (e R, M) ≅ M e$ .
Deduce $(21.7)$ : ${End}_{R} (e R) ≅ e R e$ as rings, and check the multiplication convention.
Match direct decompositions of a module with complete orthogonal families in $End (M)$ .
Prove the equivalence of the four conditions in $(21.8)$ defining a primitive idempotent.
State $(21.9)$ and explain why a local idempotent is primitive but not conversely.
Compute $Hom$ groups between the indecomposable summands of a triangular matrix ring.

Definitions

Definition(21.8)Primitive idempotent

A nonzero idempotent $e \in R$ is primitive if it admits no decomposition $e = α + β$ with $α, β$ nonzero orthogonal idempotents of $R$ . Proposition $(21.8)$ below shows this is equivalent to indecomposability of $e R$ as a right $R$ -module, to indecomposability of $R e$ as a left $R$ -module, and to the absence of nontrivial idempotents in $e R e$ .

Definition(21.9)Local idempotent

A nonzero idempotent $e \in R$ is local if the corner ring $e R e$ is a local ring, that is, a nonzero ring whose non-units form a two-sided ideal — equivalently, $e R e / rad (e R e)$ is a division ring.

Indecomposable module: A nonzero module $M$ admitting no decomposition $M = M_{1} \oplus M_{2}$ with both summands nonzero.
Strongly indecomposable: ${End}_{R} (M)$ is a local ring. This implies indecomposable, because a local ring has no idempotents besides $0$ and $1$ .
Trivial idempotents: In a ring $S$ with identity $1_{S}$ , the elements $0$ and $1_{S}$ . In the corner $e R e$ the identity is $e$ , so *nontrivial idempotent of $e R e$ * means an idempotent other than $0$ and $e$ .
Projection: An idempotent element of ${End}_{R} (M)$ ; its image and kernel are complementary submodules of $M$ .

Local rings are nonzero by convention, so the zero idempotent is neither primitive nor local. Every statement below assumes e is nonzero.

Core Concepts

Decompositions are idempotents

Let $M$ be any right $R$ -module and $E = {End}_{R} (M)$ . If $M = M_{1} \oplus M_{2}$ , the projection $π$ onto $M_{1}$ along $M_{2}$ satisfies $π^{2} = π$ in $E$ ; conversely an idempotent $π \in E$ gives $M = π (M) \oplus (1 - π) (M)$ . The correspondence is bijective and carries finite decompositions to complete orthogonal families.

M = M_{1} \oplus \dots \oplus M_{n} ⟷ 1 = π_{1} + \dots + π_{n}, π_{i} π_{j} = δ_{i j} π_{i} in {End}_{R} (M) .

(21.6a)

Direct decompositions of $M$ correspond exactly to complete orthogonal families of idempotents in ${End}_{R} (M)$ , with $M_{i} = π_{i} (M)$ .

Taking $M = R_{R}$ and using ${End}_{R} (R_{R}) ≅ R$ (the case $e = 1$ of $(21.7)$ ) recovers the familiar statement that decompositions of $R$ into right ideals correspond to complete orthogonal families of idempotents *in $R$ itself*. This is why idempotent arithmetic is inescapable in ring theory.

Computing Hom out of a summand of a free module

The universal property of the free module gives ${Hom}_{R} (R, M) ≅ M$ by $θ \mapsto θ (1)$ . Restricting along the split inclusion $e R ↪ R$ cuts this down: a map out of $e R$ is a map out of $R$ that kills $(1 - e) R$ , and the corresponding element of $M$ is then killed by $1 - e$ , that is, it lies in $M e$ . That is the entire content of $(21.6)$ , and it is why the isomorphism is natural.

Two grades of indecomposability

Indecomposable says $End (M)$ has no nontrivial idempotent. Strongly indecomposable says $End (M)$ is local — a strictly stronger condition, since $ℤ$ has no nontrivial idempotents but is not local. The gap closes for modules of finite length: by Fitting's Lemma, a finite-length module is indecomposable if and only if its endomorphism ring is local. So for artinian algebras the notions of primitive and local idempotent coincide, and for general rings they do not.

Key Results

Proposition(21.6)Evaluation at an idempotent

Let $R$ be a ring with identity, $e \in R$ an idempotent and $M$ a right $R$ -module. Then $λ (θ) = θ (e)$ defines an isomorphism of additive groups

λ : {Hom}_{R} (e R, M) ⟶ M e,

natural in $M$ . In particular, for idempotents $e, e' \in R$ there is a natural isomorphism ${Hom}_{R} (e R, e' R) ≅ e' R e$ .

Proof

** $λ$ lands in $M e$ .** For $θ \in {Hom}_{R} (e R, M)$ put $m = θ (e)$ . Since $e$ is idempotent, $m e = θ (e) e = θ (e \cdot e) = θ (e) = m$ , so $m = m e \in M e$ .

Additive and injective. Additivity is immediate. If $θ (e) = 0$ then for every $r \in R$ we get $θ (e r) = θ (e) r = 0$ , so $θ = 0$ on all of $e R$ .

Surjective. Given $m \in M e$ , define $θ (e r) = m r$ for $r \in R$ . This is well defined: if $e r = e r'$ then $e (r - r') = 0$ , and writing $m = m_{0} e$ gives $m (r - r') = m_{0} e (r - r') = 0$ . The map is clearly additive and $R$ -linear, and $λ (θ) = θ (e) = m \cdot 1 = m$ .

Naturality. For $ϕ : M \to N$ we have $λ (ϕ \circ θ) = ϕ (θ (e)) = ϕ (λ (θ))$ , and $ϕ (M e) \subseteq N e$ . Setting $M = e' R$ gives $M e = e' R e$ .

Corollary(21.7)The endomorphism ring of eR

For any idempotent $e$ in a ring $R$ , evaluation at $e$ is a ring isomorphism ${End}_{R} (e R) ≅ e R e$ .

Proof

By $(21.6)$ with $M = e R$ , $λ$ is an additive bijection onto $e R e$ . It remains to check multiplicativity. Let $θ, θ' \in {End}_{R} (e R)$ and put $m' = θ' (e) \in e R$ , so $m' = e m'$ . Then

λ (θ \circ θ') = θ (θ' (e)) = θ (e \cdot m') = θ (e) m' = λ (θ) λ (θ'),

using $R$ -linearity of $θ$ in the third step. Finally $λ (id) = e$ , the identity of $e R e$ .

Proposition(21.8)Primitivity

Let $e \neq 0$ be an idempotent of a ring $R$ . The following are equivalent:

$e R$ is indecomposable as a right $R$ -module;
$R e$ is indecomposable as a left $R$ -module;
the ring $e R e$ has no idempotents other than $0$ and $e$ ;
$e$ cannot be written as $α + β$ with $α, β$ nonzero orthogonal idempotents of $R$ .

An idempotent satisfying these conditions is called primitive. The list is left-right symmetric, so primitivity is a side-free notion.

Proof

**(1) $\Leftrightarrow$ (3).** A module is decomposable exactly when its endomorphism ring has an idempotent other than $0$ and $1$ , by the correspondence $(21.6 a)$ . By $(21.7)$ , ${End}_{R} (e R) ≅ e R e$ , whose identity is $e$ .

**(4) $\Rightarrow$ (3).** Contrapositive. Suppose $α \in e R e$ is idempotent with $α \neq 0, e$ . Let $β = e - α$ , the complementary idempotent computed *inside the ring $e R e$ *. Since $α e = α = e α$ by $(21.4)$ , we get $β^{2} = e - 2 α + α^{2} = e - α = β$ and $α β = α e - α^{2} = 0 = β α$ . Both are nonzero ( $β = 0$ would force $α = e$ ), and $e = α + β$ , contradicting (4).

**(3) $\Rightarrow$ (4).** Contrapositive again. Suppose $e = α + β$ with $α, β$ nonzero orthogonal idempotents. Then $e α = α^{2} + β α = α$ and $α e = α^{2} + α β = α$ , so $α \in e R e$ by the criterion $(21.4)$ . It is idempotent, nonzero, and different from $e$ because $β \neq 0$ . So $e R e$ has a nontrivial idempotent.

**(2) $\Leftrightarrow$ (3).** The left-module analogue of $(21.7)$ identifies ${End}_{R} (_{R} R e)$ with $e R e$ up to passing to the opposite ring, depending on which side endomorphisms are written. Either way the set of idempotents is unchanged, since $x^{2} = x$ is a condition invariant under $R ⇝ R^{op}$ . Hence $R e$ is indecomposable if and only if $e R e$ has only trivial idempotents.

Proposition(21.9)Local idempotents

Let $e \neq 0$ be an idempotent of a ring $R$ . The following are equivalent:

$e R$ is strongly indecomposable as a right $R$ -module, i.e. ${End}_{R} (e R)$ is local;
$R e$ is strongly indecomposable as a left $R$ -module;
$e R e$ is a local ring.

Such an $e$ is called a local idempotent. Every local idempotent is primitive.

Proof

(1) $\Leftrightarrow$ (3) is $(21.7)$ : the endomorphism ring is $e R e$ . (2) $\Leftrightarrow$ (3) follows by left-right symmetry, using that a ring is local if and only if its opposite is (the non-units form a two-sided ideal, a side-symmetric condition).

For the last sentence: if $S$ is local and $x \in S$ is idempotent with $x \neq 0, 1$ , then neither $x$ nor $1 - x$ is a unit — a unit idempotent equals $1$ — so both lie in the ideal of non-units, whence $1 = x + (1 - x)$ does too, a contradiction. So $e R e$ local forces $e R e$ to have only trivial idempotents, which is $(21.8) (3)$ .

Remark(21.9a)The converse fails

Primitive does not imply local. Take $R = ℤ$ and $e = 1$ : the only idempotents are $0$ and $1$ , so $1$ is primitive, but $e R e = ℤ$ is not local. By Fitting's Lemma the implication does reverse for modules of finite length, so over a left artinian ring — indeed whenever $e R$ has finite length — primitive and local idempotents are the same thing.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Evaluate at the generator

A cyclic projective module $e R$ is generated by $e$ , so a homomorphism is pinned down by one value. Every Hom computation in this section is this observation plus bookkeeping about which side the idempotent lands on.

Move 2

Transfer along End

Restate a module-theoretic property as a ring-theoretic property of $End (M)$ , then compute that ring. Indecomposable becomes no nontrivial idempotent; strongly indecomposable becomes local.

Move 3

Complement inside the corner

Given an idempotent $α$ of $e R e$ , its complement is $e - α$ , not $1 - α$ . Forgetting that the corner has its own identity is the standard slip in these arguments.

The criterion $(21.4)$ , $e R e = {r : e r = r = r e}$ , is what makes Move 3 rigorous: to show that a summand $α$ of $e$ lives in the corner you verify two equations rather than exhibiting $α$ as a product.

One further habit is worth acquiring: whenever a result about $e R$ is proved, ask whether the mirror statement about $R e$ needs a separate argument. Here it never does, because both are governed by the same ring $e R e$ — but that is a fact about these properties, not a general licence.

Worked Example

Upper triangular 2 x 2 matrices

Let $k$ be a field and $R = {(\begin{matrix} a & b \\ 0 & c \end{matrix}) : a, b, c \in k}$ , with $e = e_{11}$ and $f = e_{22} = 1 - e$ . Then

e R = (\begin{matrix} k & k \\ 0 & 0 \end{matrix}), f R = (\begin{matrix} 0 & 0 \\ 0 & k \end{matrix}), R e = (\begin{matrix} k & 0 \\ 0 & 0 \end{matrix}), R f = (\begin{matrix} 0 & k \\ 0 & k \end{matrix}) .

(E.1)

$\dim_{k} e R = 2$ , $\dim_{k} f R = 1$ ; $R = e R \oplus f R$ has total dimension $3$ , as it must.

The corners are $e R e = k e_{11} ≅ k$ and $f R f = k e_{22} ≅ k$ , both fields, hence local rings. By $(21.9)$ both $e$ and $f$ are local idempotents, so both are primitive and $R = e R \oplus f R$ is a decomposition into indecomposable projectives.

Now use $(21.6)$ to compute all four Hom groups without writing down a single homomorphism. The off-diagonal Peirce components are $e R f = k e_{12}$ and $f R e = 0$ , because $R$ has no nonzero entry in position $(2, 1)$ .

Hom groups between the two indecomposable summands, read off from $(21.6)$
Hom group	Equals	Dimension over $k$	Interpretation
${Hom}_{R} (e R, e R)$	$e R e$	1	Only scalars; $e R$ strongly indecomposable
${Hom}_{R} (f R, f R)$	$f R f$	1	Only scalars; $f R$ strongly indecomposable
${Hom}_{R} (f R, e R)$	$e R f$	1	Nonzero: $f R$ embeds as the socle of $e R$
${Hom}_{R} (e R, f R)$	$f R e$	0	No nonzero maps in this direction

The asymmetry is real and is worth checking by hand: the map $e R \to f R$ would have to send the generator $e_{11}$ into $(f R) e_{11} = 0$ . Conversely $e_{12} \in e R f$ corresponds to the map $f R \to e R$ , $e_{22} \mapsto e_{12}$ , whose image $k e_{12} = rad R$ is the unique simple submodule of $e R$ . In particular $e R not ≅ f R$ , which also follows from the dimension count.

Primitive without being local

In $R = ℤ$ the only idempotents are $0$ and $1$ , so $e = 1$ is primitive; but $e R e = ℤ$ is not local, so $1$ is not a local idempotent — equivalently $ℤ$ is indecomposable as a module over itself but not strongly indecomposable. The same phenomenon occurs for $e = 1$ in any commutative domain that is not local, for instance $k [x]$ .

Process and Workflow

Locate an idempotentEither given, or produced by lifting from

R / rad R

, or read off a matrix or quiver presentation.

Form the corner $e R e$ Use

(21.4)

: it is the set of

r

with

e r = r = r e

. For a matrix ring this is a block; for a path algebra it is the paths from a vertex to itself.

Test $e R e$ for idempotentsNo nontrivial ones means

e

is primitive and

e R

is indecomposable.

Test $e R e$ for localityIf the non-units of

e R e

form an ideal,

e

is local and

e R

is strongly indecomposable, so Krull–Schmidt style uniqueness becomes available.

Refine or stopIf

e R e

has a nontrivial idempotent

α

, replace

e

by the pair

α

e - α

and repeat on each piece.

You want to decompose $e R$ further. What do you know about $e R e$ ?

It is a division ringThen

e R

has no nontrivial endomorphisms at all;

e

is local and, if

R

is semiprime, right irreducible. Stop.

It is local but not a division ring

e R

is strongly indecomposable. Stop refining, but expect

e R

to have a nontrivial radical and a unique maximal submodule

e rad R

It has a nontrivial idempotent $α$ Split:

e = α + (e - α)

and

e R = α R \oplus (e - α) R

. Recurse into both corners.

You cannot tellPass to

\bar{R} = R / rad R

, where

\bar{e} \bar{R} \bar{e} ≅ e R e / rad (e R e)

is semisimple and idempotents are visible; then lift if the radical permits.

Comparison and Classification

Does the property naming each row force the property naming each column, for a nonzero idempotent $e$ ?
	$e R e$ has only trivial idempotents	$e R e$ local	$e R e$ a division ring	$e R$ indecomposable
Primitive	yes	no	no	yes
Local	yes	yes	no	yes
Right irreducible	yes	yes	yes	yes
Central	no	no	no	no

Does the property naming each row force the property naming each column, for a nonzero idempotent $e$ ?

Idempotents in familiar rings
Ring	Idempotent $e$	$e R e$	Primitive?	Local?
$ℤ$	$1$	$ℤ$	yes	no
$k [[x]]$	$1$	$k [[x]]$	yes	yes
$M_{n} (k)$	$e_{11}$	$k$	yes	yes
$M_{n} (k)$	$e_{11} + e_{22}$ , $n \geq 3$	$M_{2} (k)$	no	no
$T_{2} (k)$ upper triangular	$e_{11}$	$k$	yes	yes
$ℤ / 6 ℤ$	$3$	$ℤ / 2 ℤ$	yes	yes
$k \times k$	$(1, 1)$	$k \times k$	no	no

Relationship Map

e R e

division ring

e R e

local

e R e

has no nontrivial idempotent

e R

indecomposable

None of the first three arrows reverses in general. The last is an equivalence by $(21.8)$ . Over a ring for which $e R$ has finite length the middle arrow reverses as well, by Fitting's Lemma.

${Hom}_{R} (e R, M) ≅ M e$ — the engine
- specialises to
  - ${Hom}_{R} (e R, e' R) ≅ e' R e$
  - ${End}_{R} (e R) ≅ e R e$ as rings $(21.7)$
  - ${End}_{R} (R_{R}) ≅ R$ , the case $e = 1$
- yields
  - primitivity criteria $(21.8)$
  - locality criteria $(21.9)$
  - the isomorphism test for idempotents: $e R ≅ e' R$ iff $e = a b$ , $e' = b a$ for suitable $a, b$
- feeds
  - principal indecomposable modules
  - Krull–Schmidt uniqueness for finite-length modules
  - Morita equivalence via full idempotents

Applications and Industry Use

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

Representation theory

Principal indecomposables

Decomposing $R = ⨁ e_{i} R$ into indecomposables with $e_{i}$ primitive produces the projective indecomposable modules of a finite-dimensional algebra — the objects Brauer theory and Cartan matrices are built from.

Computational algebra

The MeatAxe and module splitting

Splitting a module over a finite field means producing an idempotent in its endomorphism algebra. Implementations compute $End$ by linear algebra, then look for a nontrivial idempotent; failure certifies indecomposability.

Coding theory

Idempotent generators of cyclic codes

A cyclic code of length $n$ over $𝔽_{q}$ with $\gcd (n, q) = 1$ is an ideal of $𝔽_{q} [x] / (x^{n} - 1)$ , and each is generated by a unique idempotent. Decomposing the algebra into minimal ideals decomposes the code into minimal cyclic codes.

Operator algebras

Projections and corners

For a projection $p$ in a von Neumann algebra, $End$ of the corresponding module is $p A p$ ; comparison theory of projections is the analytic analogue of the primitive-idempotent calculus.

In each case the point is the same: the object of interest is a module, but the computation happens in a ring of the form $e R e$ , where the tools are ring-theoretic.

Design Considerations

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

Which idempotents to name. A presentation of an algebra is far more usable if a complete orthogonal family of primitive idempotents is given explicitly; everything else can then be read off as a block.
How far to refine. Refining past primitive idempotents is impossible, and refining past local ones is often undesirable: locality is what buys uniqueness of decompositions.
Which side. $(21.6)$ has a mirror image for left modules with ${Hom}_{R} (R e, M) ≅ e M$ . Both are available; pick one and state it in your conventions section.
Composition convention. Writing endomorphisms on the left of module elements makes $(21.7)$ an isomorphism; writing them on the right turns it into an anti-isomorphism onto $e R e$ . Neither is wrong, but mixing them silently produces a genuine sign of error in Morita arguments.
Finite length or not. If your modules have finite length you may use primitive and local interchangeably. If not, you must track which one your theorem needs.

Computational Notes

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

For a finite-dimensional algebra $A$ over a field $k$ with $\dim_{k} A = n$ and a module $M$ with $\dim_{k} M = d$ :

${End}_{A} (M)$ is the solution space of a linear system in $d^{2}$ unknowns with $n d^{2}$ equations; solving it costs $O (n d^{4})$ field operations by naive elimination and is usually the dominant step.
Deciding indecomposability then means deciding whether ${End}_{A} (M)$ has a nontrivial idempotent. Over a finite field one computes $rad {End}_{A} (M)$ , splits the semisimple quotient, and lifts idempotents through the nilpotent radical — all polynomial time.
Over $ℚ$ the same outline works but coefficient growth dominates; implementations reduce modulo a good prime and lift.
${Hom}_{A} (e A, M) ≅ M e$ turns a linear-algebra problem into a multiplication: computing $M e$ costs one matrix product, not a nullspace calculation. Using the idempotent description where available is the single largest practical saving in this area.

Failure Modes and Common Mistakes

Do not assume a primitive idempotent is unique, or unique up to conjugacy — uniqueness statements require Krull–Schmidt hypotheses.
Do not assume $e$ primitive implies $\bar{e}$ primitive in $R / I$ for arbitrary $I$ ; that needs $I \subseteq rad R$ .
Do not confuse primitive idempotent with primitive ideal or primitive ring; the words are unrelated.
Do not apply $(21.6)$ to a non-idempotent generator: the well-definedness step uses $e^{2} = e$ twice.

Best Practices

Record the corner ring $e R e$ explicitly the first time an idempotent appears; almost every later question is a question about it.
When claiming a decomposition is unique, name the theorem (Krull–Schmidt) and check its hypothesis (local endomorphism rings).
Compute Hom groups by $(21.6)$ rather than by exhibiting maps; it is faster and leaves fewer places to be wrong.
State whether homomorphisms act on the left or the right before writing any composition.

Quick Reference

Master isomorphism

{Hom}_{R} (e R, M) ≅ M e

θ \mapsto θ (e)

Between summands

{Hom}_{R} (e R, e' R) ≅ e' R e

Endomorphisms

{End}_{R} (e R) ≅ e R e

as rings

Primitive

e R e

has only the idempotents

0, e

Local

e R e

is a local ring

Implicationlocal

\Rightarrow

primitive; converse needs finite length

Decompositions

M = ⨁ M_{i} \leftrightarrow

complete orthogonal family in

End (M)

Left version

{Hom}_{R} (R e, M) ≅ e M

for left modules

M

Reference numbers in Lam, §21
Reference	Statement
(21.6)	${Hom}_{R} (e R, M) ≅ M e$ ; hence ${Hom}_{R} (e R, e' R) ≅ e' R e$
(21.7)	${End}_{R} (e R) ≅ e R e$ as rings
(21.8)	Four equivalent forms of primitivity for $e \neq 0$
(21.9)	$e R$ strongly indecomposable $iff$ $e R e$ local; defines local idempotent

Frequently Asked Questions

Why is

{Hom}_{R} (e R, e' R)

equal to

e' R e

and not

e R e'

Because the isomorphism is evaluation at the generator $e$ of the source, and the value lies in the target $e' R$ . So the value is an element $m$ with $m = e' m$ and $m = m e$ , that is, $m \in e' R e$ . The mnemonic is that composition of maps must match multiplication of these sets, and $(e'' R e') (e' R e) \subseteq e'' R e$ only typechecks with targets on the left.

Does every indecomposable module come from a primitive idempotent?

No. The correspondence covers only direct summands of free modules — that is, projective modules with a cyclic generator. A general indecomposable module need not be projective at all. What is true is that the indecomposable direct summands of $R_{R}$ are exactly the modules $e R$ for $e$ primitive.

Is a primitive idempotent unique in any sense?

Not on its own. Over a semiperfect ring, the primitive idempotents in a complete orthogonal decomposition of $1$ are unique up to reordering and conjugation by a unit — that is the Krull–Schmidt statement in idempotent form — but this needs hypotheses. In a general ring there can be many primitive idempotents generating non-isomorphic modules and no conjugacy at all.

How do I recognise that two idempotents give isomorphic modules?

$e R ≅ e' R$ as right $R$ -modules if and only if there exist $a \in e R e'$ and $b \in e' R e$ with $a b = e$ and $b a = e'$ ; equivalently, if and only if there exist $a, b \in R$ with $a b = e$ and $b a = e'$ . This is Lam's $(21.20)$ , and it is proved by feeding a mutually inverse pair of homomorphisms through $(21.6)$ .

Why insist on strongly indecomposable for local idempotents?

Because locality of $End$ , not mere indecomposability, is what powers the Krull–Schmidt theorem and the theory of principal indecomposable modules. Over rings without finiteness conditions the two notions genuinely differ, and theorems that assume one do not survive substitution of the other.

Does

(21.6)

need

M

to be projective, finitely generated, or anything else?

No. $M$ is an arbitrary right $R$ -module and $e$ an arbitrary idempotent. The proof uses only $e^{2} = e$ and $R$ -linearity. This generality is what makes the isomorphism a useful computational tool rather than a structural theorem.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §21, results (21.6)–(21.9) (pp. 321–323).
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §§7, 12 and 27.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964, Chapter III.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley-Interscience, 1981, §6.
L. H. Rowen, Ring Theory, Volume I, Academic Press, 1988, §2.7.

AI Suggested Questions

Prove that $e R ≅ e' R$ if and only if there are $a, b \in R$ with $a b = e$ and $b a = e'$ , using the isomorphism $(21.6)$ .
State and prove Fitting's Lemma, and use it to show indecomposable equals strongly indecomposable for modules of finite length.
Give an example of a ring with a primitive idempotent $e$ for which $e R e$ is local but not a division ring.
How does the Krull–Schmidt theorem fail for modules whose endomorphism rings are indecomposable but not local?
Work out the complete orthogonal family of primitive idempotents for the path algebra of a quiver with three vertices.
What is the left-module version of $(21.6)$ , and how does the composition convention change $(21.7)$ ?
For which finite-dimensional algebras are all primitive idempotents conjugate under the unit group?

Idempotents and Module Decompositions

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Decompositions are idempotents

Computing Hom out of a summand of a free module

Two grades of indecomposability

Key Results

Proof Techniques and Method

Evaluate at the generator

Transfer along End

Complement inside the corner

Worked Example

Upper triangular 2 x 2 matrices

Primitive without being local

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Principal indecomposables

The MeatAxe and module splitting

Idempotent generators of cyclic codes

Projections and corners

Design Considerations

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

Decompositions are idempotents

Computing Hom out of a summand of a free module

Two grades of indecomposability

06Key Results

07Proof Techniques and Method

Evaluate at the generator

Transfer along End

Complement inside the corner

08Worked Example

Upper triangular 2 x 2 matrices

Primitive without being local

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Principal indecomposables

The MeatAxe and module splitting

Idempotent generators of cyclic codes

Projections and corners

13Design Considerations

14Computational Notes

15Failure Modes and Common Mistakes

16Best Practices

17Quick Reference

18Frequently Asked Questions

19Related KEVOS Topics

20References

21AI 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

Computational Notes

Failure Modes and Common Mistakes

Best Practices

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions