idempotent classifications

idempotent classifications

An a

unital ring

R

, an

idempotent

e

∈

R

is called a

division idempotent

if

e

⁢

R

⁢

e

=

{

e

⁢

r

⁢

e

:

r

∈

R

}

, with the

product

of

R

, forms a

division ring

.
If instead

e

⁢

R

⁢

e

is a

local ring

– here this means a ring with a unique

maximal
ideal

𝔪

where

e

⁢

R

⁢

e

/

𝔪

a division ring – then

e

is called a

local idempotent

.

Lemma 1

.

Any

integral domain

R

has only the

trivial

idempotents

0

and

1

. In particular, every division ring has only trivial idempotents.

Proof.

Suppose

e

∈

R

with

e

≠

0

and

e

2

=

e

=

1

⁢

e

. Then by cancellation

e

=

1

.
∎

The integers are an integral domain which is not a division ring and they
serve as a counter-example to many conjectures about idempotents of general
rings as we will explore below. However, the first important result is to
show the hierarchy of idempotents.

Theorem 2

.

Every local ring

R

has only trivial idempotents

0

and

1

.

Proof.

Let

𝔪

be the unique maximal ideal of

R

. Then

𝔪

is the

Jacobson radical

of

R

. Now suppose

e

∈

𝔪

is an
idempotent. Then

1

-

e

must be

left invertible

(following the

element

characterization

of Jacobson radicals (

http://planetmath.org/JacobsonRadical

)). So there exists some

u

∈

R

such that

1

=

u

⁢

(

1

-

e

)

. However, this produces

e

=

u

⁢

(

1

-

e

)

⁢

e

=

u

⁢

(

e

-

e

2

)

=

u

⁢

(

e

-

e

)

=

0

.

Thus every non-trivial idempotent

e

∈

R

lies outside

𝔪

.
As

R

/

𝔪

is a division ring, the only idempotents are

0

and

1

.
Thus if

e

∈

R

,

e

≠

0

is an idempotent then it projects to an
idempotent of

R

/

𝔪

and as

e

∉

𝔪

it follows

e

projects onto

1

so that

e

=

1

+

z

for some

z

∈

𝔪

. As

e

2

=

e

we find

0

=

z

+

z

2

(often called an

anti-idempotent

). Once again
as

z

∈

𝔪

we know there exists a

u

∈

R

such that

1

=

u

⁢

(

1

+

z

)

and

z

=

u

⁢

(

1

+

z

)

⁢

z

=

u

⁢

(

z

+

z

2

)

=

0

so indeed

e

=

1

.
∎

Corollary 3

.

Every division idempotent is a local idempotent, and every local idempotent
is a

primitive

idempotent.

Example 4

.

Let

R

be a unital ring. Then in

M

n

⁢

(

R

)

the standard idempotents
are the matrices

E

i

⁢

i

=

[

0

⋱

1

⋱

0

]

,

1

≤

i

≤

n

.

(i)

If

R

has only trivial idempotents (i.e.:

0

and

1

) then each

E

i

⁢

i

is a primitive idempotent of

M

n

⁢

(

R

)

.

(ii)

If

R

is a local ring then each

E

i

⁢

i

is a local idempotent.

(iii)

If

R

is a division ring then each

E

i

⁢

i

is a division idempotent.

When

R

=

R

⊕

R

then (i) is not satisfied and consequently
neither are (ii) and (iii). When

R

=

Z

then (i) is satisfied but not
(ii) nor (iii). When

R

=

R

⁢

[

[

x

]

]

– the

formal power series

ring over

R

– then (i) and (ii) are satisfied but not (iii).
Finally when

R

=

R

then all three are satisfied.

A

consequence

of the

Wedderburn-Artin theorems

classifies all

Artinian

simple
rings

as

matrix rings

over a division ring. Thus the primitive idempotents of an

Artinian ring

are all local idempotents. Without the Artinian

assumption

this may fail as we have already seen with

ℤ

.

Title

idempotent classifications

Canonical name

IdempotentClassifications

Date of creation

2013-03-22 16:48:43

Last modified on

2013-03-22 16:48:43

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 16U99

Classification

msc 20M99

Defines

division idempotent

Defines

local idempotent

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