Semiperfect Rings and Decompositions of the Identity

Q: Is the decomposition of 1 unique?

Up to reordering and conjugation by a single unit of R, yes; literally, no. Conjugating any decomposition by a unit gives another one, so in M_2(Z_p) there are infinitely many. What is genuinely determined is n and the multiset of isomorphism classes of the Re_i.

Q: How do I lift a family of orthogonal idempotents rather than a single one?

Inductively. Lift the first idempotent to e_1, then replace R by the corner (1-e_1)R(1-e_1) — which is again semiperfect, with radical (1-e_1)(rad R)(1-e_1) — and lift the next. Orthogonality is built into the construction; there is no need to correct a batch afterwards.

Q: Does n equal the number of simple modules?

No, it equals the number of indecomposable summands of _R R, counted with multiplicity. For M_2(Z_p), n = 2 but there is a single simple module. The number of isomorphism classes of simples equals the number of Wedderburn factors of R.

Q: Is Re_i simple when e_i is local?

Only its top is. Re_i is indecomposable projective and Re_i/(rad R)e_i is simple, but Re_i typically has many composition factors — for T_3(k) the module Re_3 has length 3. Counting those factors is the Cartan matrix problem.

Q: Does this decompose the ring into a direct product?

No. (23.6) decomposes the regular *module*. Splitting R as a product of rings requires central idempotents, and central idempotents of R need not lift to central idempotents of R — the obstruction that block theory in §25 exists to handle.

Executive Summary

The definition of a semiperfect ring — semisimple quotient plus idempotent lifting — is a statement about a quotient. This page converts it into a statement inside $R$ itself: $R$ is semiperfect iff $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ mutually orthogonal and every corner $e_{i} R e_{i}$ a local ring.

That reformulation is what makes the class usable. It produces the indecomposable projective modules $R e_{1}, \dots, R e_{n}$ , it produces a complete list of simple modules, and it is unique up to conjugation by a unit and reordering. Everything in §§24–25 — projective covers, blocks, basic rings, Cartan matrices — is built on this single decomposition.

(23.6)The characterisation

$1 = \sum e_{i}$ Shape of the answer

LocalEach corner

e_{i} R e_{i}

UniqueUp to conjugation and order

Overview

In a semisimple ring $\bar{R}$ the regular module splits into minimal left ideals, $\bar{R} = \bar{R} {\bar{x}}_{1} \oplus \dots \oplus \bar{R} {\bar{x}}_{n}$ , and the ${\bar{x}}_{i}$ are orthogonal primitive idempotents summing to $\bar{1}$ . Semiperfectness says that picture can be pulled back to $R$ .

\bar{R}

semisimple

\bar{1} = \sum {\bar{x}}_{i}

primitive orthogonallift to

1 = \sum e_{i}

R

each

e_{i} R e_{i}

local

Two things must be checked and neither is formal. First, that the lifted idempotents can be chosen orthogonal and that they still sum to $1$ rather than merely to something congruent to $1$ . Second, that primitivity upstairs upgrades to the much stronger property of being local. Both are settled below, and the second is $(23.5)$ .

The bridge in both directions is the corner ring $e R e$ , whose radical is $e (rad R) e$ and whose semisimple quotient is the corner $\bar{e} \bar{R} \bar{e}$ . Corners let one apply the definition to a piece of the ring rather than to the whole of it.

Learning Objectives

Prove $rad (e R e) = e (rad R) e$ and identify $e R e / rad (e R e)$ with $\bar{e} \bar{R} \bar{e}$ .
Show that when $rad S = 0$ , $f S f$ is a division ring iff $S f$ is a simple left ideal.
Prove $(23.5)$ : primitive idempotents of a semiperfect ring are local.
Prove both directions of $(23.6)$ , including why the lifted idempotents sum exactly to $1$ .
State the uniqueness in $(23.7)$ and explain why only conjugacy, not equality, can be expected.
Extract the indecomposable projectives and the simple modules from a decomposition of $1$ .

Definitions

Definition—Primitive and local idempotents

A nonzero idempotent $e \in R$ is primitive if it is not the sum of two nonzero orthogonal idempotents of $R$ ; equivalently, $0$ and $e$ are the only idempotents of $e R e$ ; equivalently, $R e$ is indecomposable as a left $R$ -module. It is local if $e R e$ is a local ring.

A local ring has no idempotents besides $0$ and $1$ , so **local $\Rightarrow$ primitive** in every ring. The converse fails: in $R = ℤ$ the idempotent $1$ is primitive but $1 \cdot R \cdot 1 = ℤ$ is not local.

$\bar{R}$: The quotient $R / rad R$ ; bars over elements denote images. Note $rad \bar{R} = 0$ always.
Orthogonal family: $e_{1}, \dots, e_{n}$ with $e_{i} e_{j} = 0$ for $i \neq j$ . Together with $\sum e_{i} = 1$ this is equivalent to $R = R e_{1} \oplus \dots \oplus R e_{n}$ .
$R e$: The projective left module generated by an idempotent; every projective direct summand of $_{R} R$ has this form.
Top of a module: $M / (rad R) M$ . For $M = R e$ over a semiperfect ring this is the simple module $\bar{R} \bar{e}$ .

Core Concepts

Corner rings and the radical

For any idempotent $e$ of any ring $R$ , the corner $e R e$ is a ring with identity $e$ , and

rad (e R e) = e (rad R) e, e R e / rad (e R e) ≅ \bar{e} \bar{R} \bar{e} .

(C.1)

The corner of the quotient is the quotient of the corner.

Consequently $e$ is a local idempotent of $R$ precisely when $\bar{e} \bar{R} \bar{e}$ is a division ring — a condition tested in the semisimple ring $\bar{R}$ , where it is elementary.

Division corners detect simple left ideals

Lemma—Corners over a semiprimitive ring

Let $S$ be a ring with $rad S = 0$ and let $f = f^{2} \in S$ be nonzero. Then $f S f$ is a division ring if and only if $S f$ is a simple left $S$ -module.

Proof

( $\Leftarrow$ ) If $S f$ is simple then $f S f ≅ {End}_{S} (S f)$ is a division ring by Schur's Lemma.

( $\Rightarrow$ ) Let $0 \neq N \subseteq S f$ be a submodule; note $N = N f$ . If $f N \neq 0$ , then $f N$ is a nonzero left ideal of the division ring $f S f$ , hence $f N = f S f ∋ f$ , so $f \in N$ and $N = S f$ . If instead $f N = 0$ , take $n, n' \in N$ and write $n = s f$ ; then $n n' = s (f n') = 0$ , so $N^{2} = 0$ . A nilpotent left ideal lies in $rad S = 0$ , forcing $N = 0$ , a contradiction. Hence $S f$ has no proper nonzero submodules.

Applied to $S = \bar{R}$ , whose radical always vanishes, this says: $e$ is local $iff$ $\bar{R} \bar{e}$ is a minimal left ideal, i.e. $\bar{e}$ is a left irreducible idempotent.

Why idempotents sum exactly to one

An orthogonal lift $e_{1}, \dots, e_{n}$ of a decomposition of $\bar{1}$ gives an idempotent $e = e_{1} + \dots + e_{n}$ with $\bar{e} = \bar{1}$ , so $1 - e \in rad R$ and $e = 1 - (1 - e) \in 1 + rad R \subseteq U (R)$ . Multiplying $e^{2} = e$ on the left by $e^{- 1}$ now gives $e = 1$ . So the lifted family automatically sums to the identity rather than to something merely congruent to it.

Key Results

Proposition(23.5)Primitive implies local in a semiperfect ring

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

Proof

We use one input from the idempotents chapter, the refinement property: if idempotents lift modulo $rad R$ , then for any idempotent $g \in R$ , every decomposition $\bar{g} = {\bar{x}}_{1} + \dots + {\bar{x}}_{m}$ into mutually orthogonal idempotents of $\bar{R}$ is the image of a decomposition $g = g_{1} + \dots + g_{m}$ into mutually orthogonal idempotents of $R$ . See Primitive, Irreducible and Local Idempotents.

**Step 1: $\bar{e}$ is primitive in $\bar{R}$ .** If $\bar{e} = \bar{u} + \bar{v}$ with $\bar{u}, \bar{v}$ nonzero orthogonal idempotents, the refinement property gives orthogonal idempotents $u, v \in R$ with $e = u + v$ and $\bar{u}, \bar{v}$ as prescribed. Neither is zero, since their images are nonzero. This contradicts primitivity of $e$ .

**Step 2: $\bar{R} \bar{e}$ is a minimal left ideal.** $\bar{R}$ is semisimple, so the direct summand $\bar{R} \bar{e}$ is a finite direct sum of simple modules. If there were at least two summands, the associated projections would split $\bar{e}$ into two nonzero orthogonal idempotents, contradicting Step 1. Hence $\bar{R} \bar{e}$ is simple.

Step 3: conclude. By Schur's Lemma $\bar{e} \bar{R} \bar{e} ≅ {End}_{\bar{R}} (\bar{R} \bar{e})$ is a division ring, and by $(C . 1)$ this ring is $e R e / rad (e R e)$ . A ring whose quotient by its radical is a division ring is local, so $e R e$ is local and $e$ is a local idempotent.

Theorem(23.6)Idempotent characterisation of semiperfectness

A ring $R$ is semiperfect if and only if $1 = e_{1} + \dots + e_{n}$ for some finite family of mutually orthogonal local idempotents $e_{1}, \dots, e_{n} \in R$ .

Proof

**( $\Rightarrow$ )** Let $R$ be semiperfect. In the semisimple ring $\bar{R}$ decompose the regular module into minimal left ideals; this yields $\bar{1} = {\bar{x}}_{1} + \dots + {\bar{x}}_{n}$ with the ${\bar{x}}_{i}$ mutually orthogonal primitive idempotents. By the refinement property quoted in $(23.5)$ , applied to the idempotent $g = 1$ , there are mutually orthogonal idempotents $e_{1}, \dots, e_{n}$ of $R$ with $e_{1} + \dots + e_{n} = 1$ and ${\bar{e}}_{i} = {\bar{x}}_{i}$ . Each $\bar{R} {\bar{e}}_{i}$ is a minimal left ideal, so ${\bar{e}}_{i} \bar{R} {\bar{e}}_{i}$ is a division ring by Schur's Lemma, and $(C . 1)$ makes each $e_{i}$ local. (Even without the refinement property in the strong form, a lift summing only to some idempotent $e$ with $\bar{e} = \bar{1}$ would force $e \in 1 + rad R \subseteq U (R)$ , hence $e = 1$ .)

**( $\Leftarrow$ )** Suppose $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ orthogonal and local. Each ${\bar{e}}_{i} \bar{R} {\bar{e}}_{i} ≅ e_{i} R e_{i} / rad (e_{i} R e_{i})$ is a division ring, so by the Lemma each $\bar{R} {\bar{e}}_{i}$ is a minimal left ideal. Since $\bar{1} = {\bar{e}}_{1} + \dots + {\bar{e}}_{n}$ with the ${\bar{e}}_{i}$ orthogonal,

\bar{R} = \bar{R} {\bar{e}}_{1} \oplus \dots \oplus \bar{R} {\bar{e}}_{n}

(23.6a)

A finite direct sum of simple left ideals — so $\bar{R}$ is semisimple.

It remains to lift an arbitrary idempotent $\bar{x} \in \bar{R}$ . Compare $\bar{R} = \bar{R} \bar{x} \oplus \bar{R} (\bar{1} - \bar{x})$ with $(23.6 a)$ . Since $\bar{R}$ is semisimple, the Krull–Schmidt property holds and, after reindexing, $\bar{R} \bar{x} ≅ \bar{R} {\bar{e}}_{1} \oplus \dots \oplus \bar{R} {\bar{e}}_{r}$ and $\bar{R} (\bar{1} - \bar{x}) ≅ \bar{R} {\bar{e}}_{r + 1} \oplus \dots \oplus \bar{R} {\bar{e}}_{n}$ . Two idempotents $\bar{x}, \bar{g}$ of a ring with $\bar{R} \bar{x} ≅ \bar{R} \bar{g}$ and $\bar{R} (\bar{1} - \bar{x}) ≅ \bar{R} (\bar{1} - \bar{g})$ are conjugate by a unit; applying this with $\bar{g} = {\bar{e}}_{1} + \dots + {\bar{e}}_{r}$ gives $\bar{y} \in U (\bar{R})$ with $\bar{y} \bar{x} {\bar{y}}^{- 1} = {\bar{e}}_{1} + \dots + {\bar{e}}_{r}$ .

Lift $\bar{y}$ to any $u \in R$ ; since $\bar{u}$ is a unit of $\bar{R}$ , $u \in U (R)$ . Then $u^{- 1} (e_{1} + \dots + e_{r}) u$ is an idempotent of $R$ whose image is $\bar{x}$ . Both clauses of $(23.1)$ hold, so $R$ is semiperfect.

Remark(23.7)(1)Uniqueness of the decomposition

The decomposition in $(23.6)$ is unique up to a permutation of the $e_{i}$ and conjugation by a single unit of $R$ . In particular $n$ is an invariant of $R$ , and the multiset of isomorphism classes of the modules $R e_{i}$ is an invariant. Equality cannot be expected: conjugating any decomposition by $u \in U (R)$ produces another one.

Remark(23.7)(2)The artinian case, seen directly

If $R$ is left artinian, $(23.5)$ and $(23.6)$ can be obtained without passing to $\bar{R}$ . A primitive idempotent $e$ makes $R e$ indecomposable, and $R e$ has finite length, so by Fitting's Lemma its endomorphism ring — namely $e R e$ , with endomorphisms written on the side opposite the scalars — is local. The descending chain condition then supplies a decomposition of $_{R} R$ into indecomposable left ideals, and the corresponding idempotents are primitive, hence local.

Corollary—Projectives and simples of a semiperfect ring

Let $R$ be semiperfect with $1 = e_{1} + \dots + e_{n}$ as in $(23.6)$ . Then each $R e_{i}$ is an indecomposable projective left module with local endomorphism ring, $R = R e_{1} \oplus \dots \oplus R e_{n}$ , and

R e_{i} / (rad R) e_{i} ≅ \bar{R} {\bar{e}}_{i}

(23.6b)

The top of each principal indecomposable is simple.

is simple. Every simple left $R$ -module is isomorphic to some $\bar{R} {\bar{e}}_{i}$ , because simple modules of $R$ are exactly those of $\bar{R}$ ; in particular a semiperfect ring has only finitely many simple left modules up to isomorphism.

Proof Techniques and Method

How these proofs work, and which move is reusable.

Move 1

Descend to a corner

To study a single idempotent, replace $R$ by $e R e$ . The radical, the semisimple quotient and the lifting property all restrict, so hypotheses are inherited and the problem shrinks.

Move 2

Refine, do not lift blindly

Do not lift the members of a family independently and hope for orthogonality. Use the refinement property: split an already-lifted idempotent, one step at a time, so that orthogonality is produced rather than repaired.

Move 3

Use invertibility as glue

Units lift modulo the radical, idempotents in general do not. Any construction that can be phrased as conjugation by a unit transfers upward for free.

The proof of $(23.6)$ uses all three: Move 2 to build the family, Move 1 to certify locality one corner at a time, and Move 3 to convert a Krull–Schmidt statement in $\bar{R}$ into an actual idempotent of $R$ .

There is also a standard counting argument worth isolating. Because $rad R$ contains no nonzero idempotent, distinct orthogonal idempotents stay distinct and orthogonal in $\bar{R}$ ; a decomposition of $\bar{1}$ into $n$ pieces therefore cannot be lifted to more than $n$ pieces, which is why $n$ is well defined.

Worked Example

Upper triangular matrices

Let $k$ be a field and $R = T_{3} (k)$ the ring of upper triangular $3 \times 3$ matrices, so $\dim_{k} R = 6$ . Being finite-dimensional, $R$ is artinian and hence semiperfect. Take $e_{i} = E_{i i}$ , the diagonal matrix units. They are orthogonal and $e_{1} + e_{2} + e_{3} = 1$ .

Each corner is $e_{i} R e_{i} = k E_{i i} ≅ k$ , a field, so each $e_{i}$ is a local idempotent — a decomposition of exactly the type $(23.6)$ promises. The associated projectives are the columns:

R e_{1} = {(\begin{matrix} a \\ 0 \\ 0 \end{matrix})}, R e_{2} = {(\begin{matrix} a \\ b \\ 0 \end{matrix})}, R e_{3} = {(\begin{matrix} a \\ b \\ c \end{matrix})},

(E.1)

Column $i$ of an upper triangular matrix has entries only in rows $1, \dots, i$ ; dimensions $1, 2, 3$ summing to $6 = \dim_{k} R$ .

The radical is the strictly upper triangular part, $\bar{R} ≅ k \times k \times k$ , and the tops are $R e_{i} / (rad R) e_{i} ≅ k$ — three pairwise non-isomorphic simple modules $S_{1}, S_{2}, S_{3}$ , matching the three factors of $\bar{R}$ . Note $R e_{1} not ≅ R e_{2} not ≅ R e_{3}$ even as $k$ -spaces, so the decomposition has three distinct isomorphism types.

A decomposition with repeats

Take $R = M_{2} (ℤ_{p})$ , semiperfect by $(23.2)$ . Here $1 = E_{11} + E_{22}$ and $E_{i i} R E_{i i} ≅ ℤ_{p}$ , a local ring, so both idempotents are local. This time $R E_{11} ≅ R E_{22}$ — both are the column module $ℤ_{p}^{2}$ — so $n = 2$ but there is only one isomorphism class of principal indecomposable, and correspondingly $\bar{R} ≅ M_{2} (𝔽_{p})$ has a single simple module.

A primitive idempotent that is not local

In $R = ℤ$ the only idempotents are $0$ and $1$ , so $1$ is primitive and $1 = e_{1}$ is a decomposition into orthogonal primitive idempotents. But $1 \cdot R \cdot 1 = ℤ$ is not local, and $ℤ$ is not semiperfect. The word local in $(23.6)$ cannot be weakened to primitive.

Process and Workflow

Compute $rad R$ Everything is read off the quotient, so this comes first. For a finite-dimensional algebra use the trace form or a Meataxe-based routine.

Decompose $\bar{R}$ Apply Wedderburn–Artin to

\bar{R} = \prod_{j} M_{n_{j}} (D_{j})

and write

\bar{1}

as a sum of matrix units — these are the primitive orthogonal idempotents.

Lift one at a timeLift

{\bar{x}}_{1}

e_{1}

, then repeat inside

(1 - e_{1}) R (1 - e_{1})

. Orthogonality comes free from working in the complementary corner.

Certify localityCheck that each

e_{i} R e_{i}

has a division ring quotient. Over an artinian ring this is automatic once

{\bar{e}}_{i}

is primitive.

Read off the module theory

R e_{i}

are the indecomposable projectives; their tops are all the simple modules; multiplicities give the Cartan matrix.

You have an orthogonal decomposition $1 = f_{1} + \dots + f_{m}$ . Is it the right one?

Every $f_{i} R f_{i}$ is localYes — this is a

(23.6)

decomposition, and

R

is semiperfect. It agrees with any other up to conjugation and reordering.

Some $f_{i} R f_{i}$ is not local but has no nontrivial idempotentsThe

f_{i}

is primitive without being local, so

R

is not semiperfect. Refining further will not help.

Some $f_{i} R f_{i}$ has a nontrivial idempotentRefine: split that

f_{i}

and repeat. Termination is guaranteed only if

R

has no infinite orthogonal family of idempotents.

Comparison and Classification

Decompositions of $1$ across the hierarchy
Ring	Decomposition of $1$	Corners $e_{i} R e_{i}$	Semiperfect?
$M_{n} (D)$ , $D$ a division ring	$E_{11} + \dots + E_{n n}$	$≅ D$	yes
$T_{3} (k)$	$E_{11} + E_{22} + E_{33}$	$≅ k$	yes
$M_{2} (ℤ_{p})$	$E_{11} + E_{22}$	$≅ ℤ_{p}$ , local	yes
$k [[x]]$	$1$ alone	$≅ k [[x]]$ , local	yes
$ℤ$	$1$ alone	$≅ ℤ$ , not local	no
$ℤ$ localised away from $2, 3$	$1$ alone	domain, not local	no
$\prod_{i = 1}^{\infty} 𝔽_{2}$	no finite decomposition into primitives	$≅ 𝔽_{2}$ for each coordinate	no

Properties of a single idempotent $e$
	$e R e$ has only trivial idempotents	$e R e$ local	$R e$ indecomposable	$R e$ has simple top
$e$ primitive (general $R$ )	yes	no	yes	no
$e$ local (general $R$ )	yes	yes	yes	yes
$e$ primitive, $R$ semiperfect	yes	yes	yes	yes
$e$ primitive, $R$ left artinian	yes	yes	yes	yes

Properties of a single idempotent $e$

The single no in the second column is the whole story: outside semiperfect rings, primitivity does not deliver a local corner, and the module $R e$ can be indecomposable with a badly behaved top.

Relationship Map

$1 = e_{1} + \dots + e_{n}$ orthogonal local — the $(23.6)$ decomposition
- gives, in $R$
  - $R = R e_{1} \oplus \dots \oplus R e_{n}$ as left modules
  - each $R e_{i}$ indecomposable projective
  - $End (R e_{i}) ≅ e_{i} R e_{i}$ local
- gives, in $\bar{R}$
  - $\bar{R} = \bar{R} {\bar{e}}_{1} \oplus \dots \oplus \bar{R} {\bar{e}}_{n}$ into minimal left ideals
  - a complete list of simple modules, with repeats
  - the number of Wedderburn factors of $\bar{R}$
- is unique
  - up to permutation of the $e_{i}$
  - up to conjugation by one unit of $R$

e

local

e

primitive

R e

indecomposable

Over a semiperfect ring all three conditions in the chain are equivalent; over an arbitrary ring only the displayed implications hold. The reverse implications are supplied by $(23.5)$ and by Fitting's Lemma in the finite-length case — see Strongly Indecomposable Modules and Local Endomorphism Rings and The Krull–Schmidt Theorem.

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

For $k G$ with $char k ∣ | G |$ , the $R e_{i}$ are the projective indecomposable modules; the Cartan matrix records the multiplicities of simples in their composition series and is the primary numerical invariant of a block.

Quivers

Vertices from idempotents

For a basic finite-dimensional algebra the $e_{i}$ are precisely the vertices of the associated quiver, and $\dim e_{i} (rad R / {rad}^{2} R) e_{j}$ counts the arrows from $j$ to $i$ . Computer algebra systems store algebras in exactly this form.

Homological algebra

Minimal projective resolutions

Each syzygy is covered by a direct sum of the $R e_{i}$ with multiplicities read off from tops. Without a $(23.6)$ decomposition there is no canonical choice, and Betti numbers cease to be well defined.

Integral lattices

Orders and reduction mod p

For an order over $ℤ_{p}$ , the decomposition of $1$ is stable under reduction, which is why the projective indecomposables of the order and of its residue algebra correspond one-to-one.

Honest summary: this decomposition is the coordinate system of the subject. It is not itself an application; it is what applications are written in.

Computational Notes

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

For a finite-dimensional algebra $A$ over a field given by structure constants, computing a full set of orthogonal primitive idempotents is standard: compute $rad A$ , decompose $A / rad A$ by Wedderburn–Artin, then lift with the Newton-style iteration $x \mapsto 3 x^{2} - 2 x^{3}$ , which doubles the precision of $x^{2} - x \in (rad A)^{t}$ at each step.
Orthogonalisation is the delicate part in practice. Implementations lift a single idempotent, then recurse into the complementary corner $(1 - e) A (1 - e)$ rather than orthogonalising a batch — the same induction as in the proof of $(23.6)$ .
Over finite fields the whole pipeline is polynomial time; over $ℚ$ the dominant cost is coefficient growth, and implementations work modulo a prime and lift.
Testing whether a computed corner $e_{i} A e_{i}$ is local reduces to testing whether its radical quotient is a division ring, which over a finite field is a matter of factoring the minimal polynomial of a random element.

Failure Modes and Common Mistakes

$n$ counts idempotents with multiplicity, not isomorphism classes of simples. In $M_{2} (ℤ_{p})$ , $n = 2$ but there is one simple module.
Infinite orthogonal decompositions are not allowed. A ring containing an infinite orthogonal family of nonzero idempotents cannot be semiperfect, since $\bar{R}$ would fail to be semisimple.
$e_{i} R e_{i}$ local does not make $R e_{i}$ simple. Its top is simple; $R e_{i}$ itself typically has several composition factors, and counting them is exactly the Cartan matrix problem.
Central idempotents are a different decomposition. $(23.6)$ decomposes the module $_{R} R$ , not the ring; ring-direct-factor decompositions require central idempotents and are the subject of block theory.

Best Practices

Fix one decomposition $1 = e_{1} + \dots + e_{n}$ at the start of an argument and keep it; re-deriving it midway invites the conjugation ambiguity to reappear.
Group the $e_{i}$ by isomorphism class of $R e_{i}$ before doing anything numerical; multiplicities and classes play different roles.
When you need locality, quote $(23.5)$ explicitly rather than relying on the reader to assume semiperfectness.
Verify a computed decomposition by checking $\sum e_{i} = 1$ , $e_{i} e_{j} = 0$ for $i \neq j$ , and $\dim R e_{i}$ summing correctly — cheap tests that catch most errors.

Quick Reference

Characterisation

R

semiperfect

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

, orthogonal, each

e_{i} R e_{i}

local

$(23.5)$ In a semiperfect ring, primitive

\Rightarrow

local

Corner radical

rad (e R e) = e (rad R) e

Locality test

e

local

iff \bar{R} \bar{e}

is a minimal left ideal

Uniquenessup to permutation and conjugation by one unit

Projectives

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

, indecomposable with local endomorphism rings

Simples

R e_{i} / (rad R) e_{i}

; finitely many up to isomorphism

Watch out

n

counts multiplicities, not isomorphism classes

Results of this page
Reference	Statement	Hypotheses
(23.5)	Primitive idempotents are local	$R$ semiperfect
(23.6)	Semiperfect $iff$ $1$ is a finite sum of orthogonal local idempotents	none beyond a ring with $1$
(23.7)(1)	Uniqueness up to conjugation and permutation	$R$ semiperfect
(23.7)(2)	Direct route via Fitting's Lemma	$R$ left artinian

Frequently Asked Questions

Why does

(23.6)

insist on local rather than primitive idempotents?

Because primitivity is too weak outside the artinian world. $ℤ$ has the decomposition $1 = 1$ into a single primitive idempotent, yet it is not semiperfect. Locality of $e_{i} R e_{i}$ is what forces $\bar{R} {\bar{e}}_{i}$ to be a minimal left ideal and hence $\bar{R}$ to be semisimple.

Is the decomposition of

1

unique?

Up to reordering and conjugation by a single unit of $R$ , yes; literally, no. Conjugating any decomposition by a unit gives another one, so in $M_{2} (ℤ_{p})$ there are infinitely many. What is genuinely determined is $n$ and the multiset of isomorphism classes of the $R e_{i}$ .

How do I lift a family of orthogonal idempotents rather than a single one?

Inductively. Lift the first idempotent to $e_{1}$ , then replace $R$ by the corner $(1 - e_{1}) R (1 - e_{1})$ — which is again semiperfect, with radical $(1 - e_{1}) (rad R) (1 - e_{1})$ — and lift the next. Orthogonality is built into the construction; there is no need to correct a batch afterwards.

Does

n

equal the number of simple modules?

No, it equals the number of indecomposable summands of $_{R} R$ , counted with multiplicity. For $M_{2} (ℤ_{p})$ , $n = 2$ but there is a single simple module. The number of isomorphism classes of simples equals the number of Wedderburn factors of $\bar{R}$ .

R e_{i}

simple when

e_{i}

is local?

Only its top is. $R e_{i}$ is indecomposable projective and $R e_{i} / (rad R) e_{i}$ is simple, but $R e_{i}$ typically has many composition factors — for $T_{3} (k)$ the module $R e_{3}$ has length $3$ . Counting those factors is the Cartan matrix problem.

Does this decompose the ring into a direct product?

No. $(23.6)$ decomposes the regular module. Splitting $R$ as a product of rings requires central idempotents, and central idempotents of $\bar{R}$ need not lift to central idempotents of $R$ — the obstruction that block theory in §25 exists to handle.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §23, results (23.5)–(23.7); background on idempotents in §21.
H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (1960), 466–488.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, chapters on semiperfect rings and on idempotents.
C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley-Interscience, 1962, chapters on idempotents and principal indecomposable modules.
L. H. Rowen, Ring Theory, Volume I, Academic Press, 1988, sections on idempotent lifting and semiperfect rings.

AI Suggested Questions

Prove that $rad (e R e) = e (rad R) e$ for every idempotent $e$ .
Show that if $R$ is semiperfect then so is the corner $e R e$ for every nonzero idempotent $e$ .
Give a ring with an infinite orthogonal family of nonzero idempotents and explain what fails in $(23.6)$ .
How is the Cartan matrix of a semiperfect ring defined in terms of the $R e_{i}$ ?
Prove that idempotents $e, f$ with $R e ≅ R f$ and $R (1 - e) ≅ R (1 - f)$ are conjugate by a unit.
What is the basic ring associated with a semiperfect ring, and why is it Morita equivalent to it?
For $k S_{3}$ with $char k = 3$ , compute a decomposition of $1$ into orthogonal local idempotents.

Semiperfect Rings and Idempotents

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Corner rings and the radical

Division corners detect simple left ideals

Why idempotents sum exactly to one

Key Results

Proof Techniques and Method

Descend to a corner

Refine, do not lift blindly

Use invertibility as glue

Worked Example

Upper triangular matrices

A decomposition with repeats

A primitive idempotent that is not local

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Principal indecomposables

Vertices from idempotents

Minimal projective resolutions

Orders and reduction mod p

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

Corner rings and the radical

Division corners detect simple left ideals

Why idempotents sum exactly to one

06Key Results

07Proof Techniques and Method

Descend to a corner

Refine, do not lift blindly

Use invertibility as glue

08Worked Example

Upper triangular matrices

A decomposition with repeats

A primitive idempotent that is not local

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Principal indecomposables

Vertices from idempotents

Minimal projective resolutions

Orders and reduction mod p

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

Computational Notes

Failure Modes and Common Mistakes

Best Practices

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions