Idempotents and the Peirce Decomposition

Q: Why is (21.3) only a decomposition of additive groups?

Because the summands are not closed under multiplication by arbitrary elements of R, and two of them are not even closed under multiplication by each other in a useful way. eRe eRf ⊆ eRf, so multiplication moves between summands according to the matrix rule. Only when the off-diagonal pieces vanish do the diagonal pieces become ideals, and then (21.3) upgrades to a product of rings.

Q: Is eRe the same thing as the left ideal Re?

No. Re is a left ideal of R and generally not closed under multiplication in a way that makes it a ring with identity; eRe is a ring with identity e. They are related by eRe = e(Re), and they coincide as sets exactly when fRe = 0, that is, when Re is already contained in eR.

Q: Does every ring have nontrivial idempotents?

No. Local rings have none — that is essentially the definition in the commutative case and a theorem in general — and integral domains have none. Rings with no nontrivial idempotents at all are exactly the ones whose regular module is indecomposable on both sides.

Q: How does the Peirce decomposition interact with the Jacobson radical?

Cleanly: rad(eRe) = e(rad R)e = (rad R) eRe for every idempotent e. So the radical is compatible with the grading, and passing to R/rad R commutes with taking corners. This is developed in *Corner Rings*.

Q: What happens if I use an infinite family of orthogonal idempotents?

The decomposition R = _i,j e_i R e_j requires _i e_i = 1, which for an infinite family cannot hold in a ring (a sum of infinitely many nonzero orthogonal idempotents is not an element). One instead works with an infinite family whose partial sums approximate 1 in some topology, or with rings without identity. Infinite orthogonal families do occur — they are exactly what obstructs Dedekind-finiteness — but they do not give a Peirce decomposition of R.

Q: Why does the commutative theory look so much simpler?

Because in a commutative ring every idempotent is central, so (21.5) is vacuous and (21.3) always collapses to a product of two rings. The whole subtlety of the noncommutative theory lies in idempotents that split modules without splitting the ring.

Executive Summary

An idempotent is the algebraic shadow of a projection. Given $e = e^{2}$ in a ring $R$ and its complement $f = 1 - e$ , every element $r$ can be written as $r = (e + f) r (e + f)$ and expanded into four pieces. That expansion — the Peirce decomposition — is the single most useful structural device attached to an idempotent, and it is available in any ring with identity, with no chain conditions and no commutativity.

Two of the four pieces, $e R e$ and $f R f$ , are rings in their own right; the other two are bimodules linking them. The upshot is that $R$ is a **generalised $2 \times 2$ matrix ring**, and structural questions about $R$ frequently reduce to the same questions about the smaller corners.

$e^{2} = e$ Hypothesis

4Peirce components

$e R f = f R e = 0$ Centrality test

1870Peirce, Linear Associative Algebra

Overview

In a commutative ring an idempotent $e$ splits $R$ as a direct product $R e \times R (1 - e)$ of rings, and the rings with no nontrivial idempotents are exactly the indecomposable ones. For noncommutative rings that statement survives only after inserting the word central: $R \neq 0$ decomposes as a product of two nonzero rings precisely when it has a nontrivial central idempotent. But a noncommutative ring can be indecomposable and still carry a rich supply of non-central idempotents, and those are what make the theory interesting.

A non-central idempotent does not split the ring, but it still splits the additive group, and it splits the regular module on either side. Those weaker splittings carry an enormous amount of information.

R = R e \oplus R f (left ideals), R = e R \oplus f R (right ideals),

(21.1)-(21.2)

The two one-sided Peirce decompositions, with $f = 1 - e$ .

R = e R e \oplus e R f \oplus f R e \oplus f R f (additive groups) .

(21.3)

The two-sided Peirce decomposition. Only the diagonal pieces are rings.

The corner ring $e R e$ is the object that recurs everywhere downstream: it is the endomorphism ring of the projective module $e R$ , its radical is $e (rad R) e$ , and when $e$ is full the ideal theory of $R$ and of $e R e$ agree. Those developments are taken up in Corner Rings, Idempotents and Direct Decompositions of Modules and Matrix Units, Full Idempotents and Matrix Ring Recognition.

Learning Objectives

State the three Peirce decompositions $(21.1)$ , $(21.2)$ , $(21.3)$ and say which are decompositions of modules and which merely of abelian groups.
Prove that $e R e$ is a ring with identity $e$ and that $e R e = {r \in R : e r = r = r e}$ .
Prove $(21.5)$ : $e \in Z (R)$ if and only if $e R f = f R e = 0$ .
Assemble $R$ as a generalised $2 \times 2$ matrix ring over the two corners.
Compute the four Peirce components of $M_{n} (k)$ for the block idempotent $diag (1, \dots, 1, 0, \dots, 0)$ .
Extend the decomposition to a complete orthogonal family $1 = e_{1} + \dots + e_{n}$ .

Definitions

Definition(21.0)Idempotents and complements

Let $R$ be a ring with identity. An element $e \in R$ is an idempotent if $e^{2} = e$ . The elements $0$ and $1$ are the trivial idempotents. For any idempotent $e$ , its complementary idempotent is $f = 1 - e$ ; then $f^{2} = 1 - 2 e + e^{2} = 1 - e = f$ and $e f = f e = e - e^{2} = 0$ , so ${e, f}$ is a pair of orthogonal idempotents summing to $1$ .

$e R e$: The corner ring at $e$ : the set of all products $e r e$ with $r \in R$ , closed under addition and multiplication, with multiplicative identity $e$ .
$e R f$ , $f R e$: The off-diagonal Peirce components. $e R f$ is an $(e R e, f R f)$ -bimodule and $f R e$ is an $(f R f, e R e)$ -bimodule.
Complete orthogonal family: Idempotents $e_{1}, \dots, e_{n}$ with $e_{i} e_{j} = 0$ for $i \neq j$ and $e_{1} + \dots + e_{n} = 1$ .
Indecomposable ring: A nonzero ring that is not a direct product of two nonzero rings; equivalently, one with no nontrivial central idempotent.
$Z (R)$: The centre of $R$ , the set of elements commuting with every element of $R$ .

Throughout, ring means ring with identity, and subring is not required to share that identity: the corner eRe is a subring of R only in the weaker sense, since its identity is e.

Core Concepts

Why the four pieces are independent

Write $r = 1 \cdot r \cdot 1 = (e + f) r (e + f) = e r e + e r f + f r e + f r f$ . Every element is therefore a sum of one term from each component. Independence is equally cheap: if $x \in e R e$ then $e x e = x$ , while every element of $e R f$ , $f R e$ and $f R f$ is killed on the left by $e$ or on the right by $e$ . Applying the operators $r \mapsto e r e$ , $r \mapsto e r f$ , $r \mapsto f r e$ , $r \mapsto f r f$ to a vanishing sum extracts each term in turn, so the sum in $(21.3)$ is direct.

The same argument with $1 = e + f$ inserted on one side only gives $(21.1)$ and $(21.2)$ : $r = r e + r f$ exhibits $R = R e + R f$ , and $x \in R e \cap R f$ satisfies $x = x e$ and $x e = 0$ , hence $x = 0$ .

The corners are rings, the off-diagonals are not

Multiplication respects the grading in the manner of matrix blocks: $(e R e) (e R e) \subseteq e R e$ and $(e R e) (e R f) \subseteq e R f$ , while $(e R f) (e R f) \subseteq e R f f R f = 0$ because $f e = 0$ . So $e R e$ and $f R f$ are closed under multiplication and contain $e$ and $f$ as identities, whereas $e R f$ and $f R e$ square to zero and are only bimodules.

e R e = {r \in R : e r = r = r e}, f R f = {r \in R : f r = r = r f} .

(21.4)

An intrinsic description of the corners: membership is an equation, not an existential statement.

R as a generalised matrix ring

Set $A = e R e$ , $B = f R f$ , $M = e R f$ , $N = f R e$ . Then $M$ is an $(A, B)$ -bimodule, $N$ is a $(B, A)$ -bimodule, and multiplication in $R$ supplies pairings $M \otimes_{B} N \to A$ and $N \otimes_{A} M \to B$ that are associative against the bimodule actions. Assembling the four components into a matrix reproduces multiplication in $R$ exactly.

R ≅ (\begin{matrix} e R e & e R f \\ f R e & f R f \end{matrix}), r ⟼ (\begin{matrix} e r e & e r f \\ f r e & f r f \end{matrix}),

(21.3')

An isomorphism of rings, where the right-hand side is multiplied as a matrix using the four internal products.

More than two blocks

Nothing forces the family to have two members. If $1 = e_{1} + \dots + e_{n}$ with the $e_{i}$ pairwise orthogonal idempotents, the same insertion argument yields the finer decomposition below, and $R$ becomes a generalised $n \times n$ matrix ring with diagonal entries the corners $e_{i} R e_{i}$ .

R = ⨁_{i = 1}^{n} ⨁_{j = 1}^{n} e_{i} R e_{j}, (e_{i} R e_{j}) (e_{k} R e_{l}) \subseteq δ_{j k} e_{i} R e_{l} .

(21.3'')

The general Peirce decomposition. The Kronecker delta encodes the matrix multiplication rule.

Key Results

Proposition(21.1)-(21.3)The Peirce decompositions

Let $R$ be a ring with identity, $e \in R$ an idempotent and $f = 1 - e$ . Then:

$R = R e \oplus R f$ as left $R$ -modules, and $R e$ , $R f$ are left ideals;
$R = e R \oplus f R$ as right $R$ -modules, and $e R$ , $f R$ are right ideals;
$R = e R e \oplus e R f \oplus f R e \oplus f R f$ as additive groups.

Moreover $e R e$ and $f R f$ are rings with identities $e$ and $f$ respectively, and $(e R f)^{2} = (f R e)^{2} = 0$ .

Proof

(1). For $r \in R$ , $r = r e + r f$ with $r e \in R e$ and $r f \in R f$ , so $R = R e + R f$ . If $x \in R e \cap R f$ , write $x = a e = b f$ . Then $x e = a e^{2} = a e = x$ and also $x e = b f e = 0$ , so $x = 0$ . Both $R e$ and $R f$ are visibly left ideals. (2) is the same computation on the other side.

(3). Expanding $r = (e + f) r (e + f) = e r e + e r f + f r e + f r f$ shows $R = e R e + e R f + f R e + f R f$ . For directness, suppose $x_{1} + x_{2} + x_{3} + x_{4} = 0$ with $x_{1} \in e R e$ , $x_{2} \in e R f$ , $x_{3} \in f R e$ , $x_{4} \in f R f$ . Multiply the relation on the left and on the right by $e$ . By $(21.4)$ we have $e x_{1} e = x_{1}$ ; and $e x_{2} e = 0$ because $x_{2} = x_{2} f$ and $f e = 0$ ; and $e x_{3} e = 0$ because $x_{3} = f x_{3}$ and $e f = 0$ ; and $e x_{4} e = 0$ for either reason. Hence $x_{1} = 0$ . The other three terms are killed the same way using $e □ f$ , $f □ e$ and $f □ f$ .

Ring structure. $(e r e) (e r' e) = e (r e r') e \in e R e$ , and $e (e r e) = e r e = (e r e) e$ , so $e R e$ is closed under multiplication with identity $e$ . Finally $(e r f) (e r' f) = e r (f e) r' f = 0$ since $f e = 0$ .

Lemma(21.4)Intrinsic description of a corner

For an idempotent $e$ in a ring $R$ , $e R e = {r \in R : e r = r = r e}$ .

Proof

If $r = e s e$ then $e r = e^{2} s e = e s e = r$ and likewise $r e = e s e^{2} = r$ . Conversely, if $e r = r = r e$ then $r = e r e \in e R e$ . Both inclusions use only $e^{2} = e$ .

Lemma(21.5)Centrality criterion

Let $e$ be an idempotent in $R$ and $f = 1 - e$ . Then $e \in Z (R)$ if and only if $e R f = 0$ and $f R e = 0$ .

Proof

For any $r \in R$ , insert $1 = e + f$ on the outside: $e r = e r (e + f) = e r e + e r f$ and $r e = (e + f) r e = e r e + f r e$ . Subtracting, $e r - r e = e r f - f r e$ . The two terms lie in the independent summands $e R f$ and $f R e$ of $(21.3)$ , so $e r = r e$ holds if and only if $e r f = 0$ and $f r e = 0$ . Letting $r$ range over $R$ gives the statement.

Corollary(21.5a)Central idempotents split the ring

If $e$ is a central idempotent of $R$ with $f = 1 - e$ , then $e R = e R e$ and $f R = f R f$ are two-sided ideals and $R ≅ e R e \times f R f$ as rings. Consequently a nonzero ring is indecomposable as a direct product of rings exactly when its only central idempotents are $0$ and $1$ .

Proof

By $(21.5)$ the off-diagonal components vanish, so $(21.3)$ collapses to $R = e R e \oplus f R f$ , a direct sum of two-sided ideals each of which is a ring with identity ( $e$ and $f$ ). The map $r \mapsto (e r e, f r f) = (e r, f r)$ is then a ring isomorphism, with inverse $(x, y) \mapsto x + y$ . Conversely, a decomposition $R ≅ S \times T$ with $S, T \neq 0$ makes $(1_{S}, 0)$ a nontrivial central idempotent.

Remark(21.5b)One-sided degeneracy

Vanishing of a single off-diagonal component is a genuinely weaker and useful condition: $f R e = 0$ holds if and only if $e R$ is a two-sided ideal, and then the matrix picture $(21.3')$ is upper triangular. Triangular matrix rings are the standard source of examples where exactly one of the two components dies.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Insert the resolution of identity

Replace $r$ by $(e_{1} + \dots + e_{n}) r (e_{1} + \dots + e_{n})$ and expand. This is the whole content of the Peirce decomposition and is the first line of most idempotent arguments.

Move 2

Project to isolate a component

To show a sum of graded pieces vanishes termwise, hit it with $e_{i} \cdot □ \cdot e_{j}$ . Orthogonality kills every term but one. This replaces any appeal to linear independence.

Move 3

Read equations as block conditions

Translate a hypothesis about $R$ into a statement about which blocks vanish or which block maps are surjective. Centrality, triangularity and fullness all have such readings.

Move 2 deserves emphasis because it is what makes the decomposition canonical: the four projections $r \mapsto e r e$ and so on are additive idempotent operators on $R$ , mutually orthogonal and summing to the identity operator. The Peirce decomposition is literally a resolution of the identity of $End (R, +)$ .

A recurring subtlety: these projections are not ring maps. $r \mapsto e r e$ is additive and unital onto $e R e$ , but multiplicative only when the off-diagonal blocks vanish. Arguments that silently treat $r \mapsto e r e$ as a ring homomorphism are the most common error in this area.

Worked Example

A block idempotent in a full matrix ring

Let $k$ be any ring, $R = M_{n} (k)$ , fix $1 \leq r < n$ and let $e = diag (1, \dots, 1, 0, \dots, 0)$ with $r$ ones, so $f = 1 - e = diag (0, \dots, 0, 1, \dots, 1)$ with $n - r$ ones. Multiplying on the left by $e$ keeps the first $r$ rows and zeroes the rest; multiplying on the right by $e$ keeps the first $r$ columns. Hence:

e R e = (\begin{matrix} M_{r} (k) & 0 \\ 0 & 0 \end{matrix}), e R f = (\begin{matrix} 0 & M_{r \times (n - r)} (k) \\ 0 & 0 \end{matrix}),

(E.1)

f R e = (\begin{matrix} 0 & 0 \\ M_{(n - r) \times r} (k) & 0 \end{matrix}), f R f = (\begin{matrix} 0 & 0 \\ 0 & M_{n - r} (k) \end{matrix}) .

(E.2)

The Peirce decomposition of $M_{n} (k)$ is exactly the partition of a matrix into four blocks.

So $e R e ≅ M_{r} (k)$ and $f R f ≅ M_{n - r} (k)$ as rings, and the generalised matrix ring $(21.3')$ is the honest block decomposition of $M_{n} (k)$ . Both off-diagonal blocks are nonzero whenever $k \neq 0$ and $0 < r < n$ , which by $(21.5)$ confirms that $e$ is not central — as it must not be, since $Z (M_{n} (k)) = Z (k) \cdot 1$ consists of scalar matrices.

The extreme corner

Take $r = 1$ , so $e = e_{11}$ is a matrix unit. Then $e R e$ consists of the matrices supported in position $(1, 1)$ , and $e_{11} r e_{11} = r_{11} e_{11}$ for $r = (r_{i j})$ . Thus $e_{11} M_{n} (k) e_{11} ≅ k$ : the corner at a single matrix unit recovers the coefficient ring. This is the computation behind Matrix Units, Full Idempotents and Matrix Ring Recognition.

A commutative sanity check

In $R = ℤ / 6 ℤ$ the idempotents are $0, 1, 3, 4$ , since $3^{2} = 9 = 3$ and $4^{2} = 16 = 4$ . Take $e = 3$ , $f = 4$ . Everything is central, so $e R f = f R e = 0$ and $(21.3)$ collapses to $R = 3 R \oplus 4 R$ , that is $ℤ / 6 ≅ ℤ / 2 \times ℤ / 3$ — the Chinese Remainder Theorem read as a Peirce decomposition.

Comparison and Classification

What each Peirce component is, and what structure it carries
Component	Description	Structure	Vanishes when
$e R e$	${r : e r = r = r e}$	Ring with identity $e$	$e = 0$
$e R f$	${r : e r = r = r f}$	$(e R e, f R f)$ -bimodule, square zero	$e R$ is a left ideal too
$f R e$	${r : f r = r = r e}$	$(f R f, e R e)$ -bimodule, square zero	$e R$ is a two-sided ideal
$f R f$	${r : f r = r = r f}$	Ring with identity $f$	$e = 1$

Which decomposition is a decomposition of what
	Left modules	Right modules	Rings	Abelian groups
$R = R e \oplus R f$	yes	no	no	yes
$R = e R \oplus f R$	no	yes	no	yes
$R = ⨁ e_{i} R e_{j}$	no	no	no	yes
$R = e R e \times f R f$ , $e$ central	yes	yes	yes	yes

Which decomposition is a decomposition of what

The last row is the point of the whole section: only for central $e$ does the decomposition upgrade all the way to rings. Everything below that line is a decomposition of modules or of additive groups only.

Relationship Map

The logical dependencies among the notions attached to a single idempotent form a short chain of strengthenings.

e^{2} = e

R = e R \oplus f R

{End}_{R} (e R) ≅ e R e

e R

indecomposable

\Leftrightarrow

e R e

has no nontrivial idempotent

Idempotent $e \in R$ — with complement $f = 1 - e$
- always gives
  - left ideal decomposition $(21.1)$
  - right ideal decomposition $(21.2)$
  - four-block additive decomposition $(21.3)$
  - a corner ring $e R e$ with identity $e$
- gives a ring product only if
  - $e \in Z (R)$ , equivalently $e R f = f R e = 0$ by $(21.5)$
- gives a triangular picture if
  - exactly one of $e R f$ , $f R e$ vanishes

Idempotents of $R$

e^{2} = e

; always yield a Peirce decomposition

Primitive idempotents

e R

indecomposable; equivalently

e R e

has only trivial idempotents

Local idempotents

e R e

is a local ring

Right irreducible idempotents

e R

is a minimal right ideal;

e R e

is a division ring

Central idempotents sit outside this tower rather than inside it: a central idempotent need not be primitive, and a primitive idempotent is rarely central. The two hierarchies meet only in the block-diagonal case.

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

Block decomposition of group algebras

The central primitive idempotents of $k G$ cut the algebra into blocks; the non-central primitive idempotents cut the regular module into principal indecomposables. Brauer theory is organised entirely around these two families.

Quiver algebras

Vertex idempotents

In a path algebra $k Q$ the vertices give a complete orthogonal family $1 = \sum e_{v}$ , and $e_{v} (k Q) e_{w}$ is the span of paths from $w$ to $v$ . The Peirce decomposition is the grading by source and target.

Operator algebras

Corner algebras

For a projection $p$ in a $C^{*}$ -algebra $A$ , the corner $p A p$ is again a $C^{*}$ -algebra and $A$ has the same $2 \times 2$ block picture. Full projections are the ones for which $p A p$ is Morita equivalent to $A$ .

Symbolic computation

Splitting an algebra

Computer algebra systems decompose a finite-dimensional algebra by finding an idempotent and recursing into the corners $e A e$ and $f A f$ ; the off-diagonal blocks are recovered afterwards as bimodules.

The honest summary is that the Peirce decomposition is infrastructure. It is almost never the theorem; it is the coordinate system in which the theorem is stated.

Design Considerations

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

Which idempotent? Different idempotents give different coordinate systems. Choose one whose corner $e R e$ you already understand — a matrix unit, a vertex of a quiver, a projection onto an isotypic component.
How many blocks? A complete orthogonal family of $n$ idempotents gives an $n \times n$ picture. Refining past primitive idempotents is impossible, so primitivity is the natural stopping condition.
Left or right? $(21.1)$ and $(21.2)$ are mirror images and both are always available, but downstream results are frequently one-sided. Fix a side before you start and record it.
Central or not? If you need a decomposition of rings, you must find a central idempotent; a merely non-central one buys you only modules and bimodules. Do not conflate the two.
Identity conventions. The corner $e R e$ has identity $e \neq 1$ , so it is not a unital subring of $R$ . Any functor or software interface that insists on identity-preserving inclusions will refuse this construction.

Standards and Notation

Standards here covers notation, symbol and markup standards, and reference implementations, rather than material or design codes.

Complement

f = 1 - e

(Lam);

e'

e^{⊥}

in operator-algebra sources

Corner

e R e

universally;

R_{e}

occasionally in older algebra texts

Matrix ring

M_{n} (R)

in this collection;

R^{n \times n}

and

{Mat}_{n} (R)

also occur

Matrix units

e_{i j}

with

e_{i j} e_{k l} = δ_{j k} e_{i l}

Generalised matrix ringWritten as a formal

2 \times 2

array of the two rings and two bimodules

MarkupPresentation MathML per ISO/IEC 40314; symbol conventions per ISO 80000-2

GAP / MagmaCentralIdempotentsOfAlgebra, CentralIdempotents

Failure Modes and Common Mistakes

Do not assume the four components have equal size or are isomorphic — for $e = e_{11}$ in $M_{n} (k)$ they have $k$ -dimensions $1$ , $n - 1$ , $n - 1$ , $(n - 1)^{2}$ .
Do not assume $e R f$ and $f R e$ are ideals of anything; they are bimodules over different pairs of rings.
Do not conflate $R e$ with $e R e$ . The former is a left ideal of $R$ ; the latter is a ring. They coincide only when $f R e = 0$ , and even then only as sets.
Do not expect idempotents to be conjugate. Two idempotents can generate isomorphic corners without being related by any unit of $R$ .

Best Practices

Always name the complement $f = 1 - e$ explicitly at the start; half the errors in this area come from implicit complements.
When you write a block matrix, say once whether rows are indexed by $e$ -then- $f$ or the reverse, and never change it.
Verify any claimed Peirce computation by dimension count or by checking that the four projections sum to the identity.
Before invoking a corner-ring theorem, check whether it needs $e$ to be nonzero, primitive, full or central — these are four different hypotheses.

Historical Notes and Lessons Learned

1870Peirce's Linear Associative AlgebraBenjamin Peirce circulates the memoir that introduces the words idempotent and nilpotent and exploits idempotents to decompose finite-dimensional associative algebras. It was printed for wide circulation only in 1881, with notes by his son Charles Sanders Peirce.
1907Wedderburn's structure theoryWedderburn's classification of finite-dimensional algebras uses idempotents systematically to peel off simple summands; the Peirce decomposition is the working tool.
1930s-1940sIdempotents in general ringsWith Jacobson's radical available, corner rings and their radicals can be discussed without chain conditions, and the decomposition becomes a tool for arbitrary rings.
1958Morita theoryMorita's equivalence theorem explains why full idempotents lose no information: $R$ and $e R e$ have equivalent module categories exactly when $R e R = R$ and $e R$ generates.

The methodological lesson is that a decomposition need not be a decomposition of the object in its own category to be useful. Peirce's splitting is only additive, yet it controls modules, ideals, radicals and equivalences — because the multiplication respects the grading even though it does not preserve the summands.

Quick Reference

Setup

e^{2} = e

f = 1 - e

e f = f e = 0

Left form

R = R e \oplus R f

(left ideals)

Right form

R = e R \oplus f R

(right ideals)

Two-sided form

R = e R e \oplus e R f \oplus f R e \oplus f R f

Corner test

r \in e R e iff e r = r = r e

Centrality

e \in Z (R) iff e R f = f R e = 0

Matrix picture

R ≅ (\begin{matrix} e R e & e R f \\ f R e & f R f \end{matrix})

Square-zero

(e R f)^{2} = (f R e)^{2} = 0

Reference numbers in Lam, §21
Reference	Statement
(21.1)	$R = R e \oplus R f$ , a decomposition into left ideals
(21.2)	$R = e R \oplus f R$ , a decomposition into right ideals
(21.3)	$R = e R e \oplus e R f \oplus f R e \oplus f R f$ , additive
(21.4)	$e R e = {r : e r = r = r e}$ and its mirror for $f$
(21.5)	$e$ central $iff$ $e R f = f R e = 0$

Frequently Asked Questions

Why is

(21.3)

only a decomposition of additive groups?

Because the summands are not closed under multiplication by arbitrary elements of $R$ , and two of them are not even closed under multiplication by each other in a useful way. $e R e \cdot e R f \subseteq e R f$ , so multiplication moves between summands according to the matrix rule. Only when the off-diagonal pieces vanish do the diagonal pieces become ideals, and then $(21.3)$ upgrades to a product of rings.

e R e

the same thing as the left ideal

R e

No. $R e$ is a left ideal of $R$ and generally not closed under multiplication in a way that makes it a ring with identity; $e R e$ is a ring with identity $e$ . They are related by $e R e = e (R e)$ , and they coincide as sets exactly when $f R e = 0$ , that is, when $R e$ is already contained in $e R$ .

Does every ring have nontrivial idempotents?

No. Local rings have none — that is essentially the definition in the commutative case and a theorem in general — and integral domains have none. Rings with no nontrivial idempotents at all are exactly the ones whose regular module is indecomposable on both sides.

How does the Peirce decomposition interact with the Jacobson radical?

Cleanly: $rad (e R e) = e (rad R) e = (rad R) \cap e R e$ for every idempotent $e$ . So the radical is compatible with the grading, and passing to $R / rad R$ commutes with taking corners. This is developed in Corner Rings.

What happens if I use an infinite family of orthogonal idempotents?

The decomposition $R = ⨁_{i, j} e_{i} R e_{j}$ requires $\sum_{i} e_{i} = 1$ , which for an infinite family cannot hold in a ring (a sum of infinitely many nonzero orthogonal idempotents is not an element). One instead works with an infinite family whose partial sums approximate $1$ in some topology, or with rings without identity. Infinite orthogonal families do occur — they are exactly what obstructs Dedekind-finiteness — but they do not give a Peirce decomposition of $R$ .

Why does the commutative theory look so much simpler?

Because in a commutative ring every idempotent is central, so $(21.5)$ is vacuous and $(21.3)$ always collapses to a product of two rings. The whole subtlety of the noncommutative theory lies in idempotents that split modules without splitting the ring.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §21, results (21.1)–(21.5) (pp. 318–321).
B. Peirce, “Linear associative algebra”, American Journal of Mathematics 4 (1881), 97–229, with notes by C. S. Peirce; first circulated in lithograph in 1870.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964, Chapter III.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §7 and §21.
L. H. Rowen, Ring Theory, Volume I, Academic Press, 1988, §1.1 and §2.7.

AI Suggested Questions

Show that the projections $r \mapsto e_{i} r e_{j}$ are mutually orthogonal additive idempotent operators summing to the identity of $End (R, +)$ .
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Compute the Peirce decomposition of the upper triangular ring $T_{2} (k)$ at each of its four idempotents.
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Idempotents and Peirce Decomposition

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Why the four pieces are independent

The corners are rings, the off-diagonals are not

R as a generalised matrix ring

More than two blocks

Key Results

Proof Techniques and Method

Insert the resolution of identity

Project to isolate a component

Read equations as block conditions

Worked Example

A block idempotent in a full matrix ring

The extreme corner

A commutative sanity check

Comparison and Classification

Relationship Map

Applications and Industry Use

Block decomposition of group algebras

Vertex idempotents

Corner algebras

Splitting an algebra

Design Considerations

Standards and Notation

Failure Modes and Common Mistakes

Best Practices

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Why the four pieces are independent

The corners are rings, the off-diagonals are not

R as a generalised matrix ring

More than two blocks

06Key Results

07Proof Techniques and Method

Insert the resolution of identity

Project to isolate a component

Read equations as block conditions

08Worked Example

A block idempotent in a full matrix ring

The extreme corner

A commutative sanity check

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Block decomposition of group algebras

Vertex idempotents

Corner algebras

Splitting an algebra

12Design Considerations

13Standards and Notation

14Failure Modes and Common Mistakes

15Best Practices

16Historical Notes and Lessons Learned

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

Comparison and Classification

Relationship Map

Applications and Industry Use

Design Considerations

Standards and Notation

Failure Modes and Common Mistakes

Best Practices

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions