Fitting’s lemma

Fitting’s lemma

Theorem 1

(Fitting Decomposition Theorem).

Let

R

be a ring, and

M

a

finite-length module

over

R

. Then for any

ϕ

∈

End

⁡

(

M

)

, the

endomorphism ring

of

M

, there is a positive integer

n

such that

M

=

ker

⁡

(

ϕ

n

)

⊕

im

⁡

(

ϕ

n

)

.

Proof.

Given

ϕ

∈

End

⁡

(

M

)

, it is clear that

ker

⁡

(

ϕ

i

)

⊆

ker

⁡

(

ϕ

i

+

1

)

and

im

⁡

(

ϕ

i

)

⊇

im

⁡

(

ϕ

i

+

1

)

for any positive integer

i

. Therefore, we have an ascending chain of

submodules

ker

⁡

(

ϕ

)

⊆

⋯

⊆

ker

⁡

(

ϕ

i

)

⊆

ker

⁡

(

ϕ

i

+

1

)

⊆

⋯

,

and a descending chain of submodules

im

⁡

(

ϕ

)

⊇

⋯

⊇

im

⁡

(

ϕ

i

)

⊇

im

⁡

(

ϕ

i

+

1

)

⊇

⋯

.

Both chains must be finite, since

M

has finite length. Therefore, we can find a positive integer

n

such that

{

ker

⁡

(

ϕ

n

)

=

ker

⁡

(

ϕ

n

+

1

)

=

⋯

,

and

im

⁡

(

ϕ

n

)

=

im

⁡

(

ϕ

n

+

1

)

=

⋯

.

If

u

∈

M

, then

ϕ

n

⁢

(

u

)

∈

im

⁡

(

ϕ

n

)

=

im

⁡

(

ϕ

2

⁢

n

)

. Therefore,

ϕ

n

⁢

(

u

)

=

ϕ

2

⁢

n

⁢

(

v

)

for some

v

∈

M

. Write

u

=

(

u

-

ϕ

n

⁢

(

v

)

)

+

ϕ

n

⁢

(

v

)

. Applying the

ϕ

n

to the first term, we get

ϕ

n

⁢

(

u

-

ϕ

n

⁢

(

v

)

)

=

ϕ

n

⁢

(

u

)

-

ϕ

2

⁢

n

⁢

(

v

)

=

0

, so it is in

ker

⁡

(

ϕ

n

)

. The second term is clearly in

im

⁡

(

ϕ

n

)

. So

M

=

ker

⁡

(

ϕ

n

)

+

im

⁡

(

ϕ

n

)

.

Furthermore, if

u

∈

ker

⁡

(

ϕ

n

)

∩

im

⁡

(

ϕ

n

)

, then

u

=

ϕ

n

⁢

(

v

)

for some

v

∈

M

. Since

ϕ

2

⁢

n

⁢

(

v

)

=

ϕ

n

⁢

(

u

)

=

0

,

v

∈

ker

⁡

(

ϕ

2

⁢

n

)

=

ker

⁡

(

ϕ

n

)

. Therefore,

u

=

ϕ

n

⁢

(

v

)

=

0

. This shows that we can replace

+

in the

equation

above by

⊕

, proving the

theorem

.
∎

Stated differently, the theorem says that, given an

endomorphism

ϕ

on

M

,

M

can be decomposed into two submodules

M

1

and

M

2

, such that

ϕ

restricted to

M

1

is

nilpotent

, and

ϕ

restricted to

M

2

is an

isomorphism

.

A direct

consequence

of this decomposition property is the famous

Fitting Lemma

:

Corollary 1

(Fitting Lemma).

In the theorem above,

ϕ

is either nilpotent (

ϕ

n

=

0

for some

n

) or an automorphism iff

M

is

indecomposable

.

Proof.

Suppose first that

M

is indecomposable. Then either

ker

⁡

(

ϕ

n

)

=

0

or

im

⁡

(

ϕ

n

)

=

0

. If

n

=

1

, then the lemma is proved. Suppose

n

>

1

. In the former case, any

u

∈

M

is the image of some

v

under

ϕ

n

, so

u

=

ϕ

⁢

(

ϕ

n

-

1

⁢

(

v

)

)

and therefore

ϕ

is onto. If

ϕ

⁢

(

u

)

=

0

, then

ϕ

n

⁢

(

u

)

=

ϕ

n

-

1

⁢

(

ϕ

⁢

(

u

)

)

=

0

, so

u

=

0

. This means

u

is an automorphism. In the latter case,

ϕ

n

⁢

(

u

)

=

0

for any

u

∈

M

, so

ϕ

is nilpotent.

Now suppose

M

is not indecomposable. Then writing

M

=

M

1

⊕

M

2

, where

M

1

and

M

2

as

proper submodules

of

M

, we can define

ϕ

∈

End

⁡

(

M

)

such that

ϕ

is the

identity

on

M

1

and

0

on

M

2

(

ϕ

is a

projection

of

M

onto

M

1

). Since both

M

1

and

M

2

are proper,

ϕ

is neither an automorphism nor nilpotent.
∎

Remark

. Another way of stating Fitting Lemma is to say that

End

⁡

(

M

)

is a

local ring

iff the finite-length module

M

is indecomposable. The (unique)

maximal ideal

in

End

⁡

(

M

)

consists of all nilpotent endomorphisms (and its

complement

consists of, of course, the automorphisms).

Title

Fitting’s lemma

Canonical name

FittingsLemma

Date of creation

2013-03-22 17:29:26

Last modified on

2013-03-22 17:29:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 16D10

Classification

msc 16S50

Classification

msc 13C15

Synonym

Fitting lemma

Synonym

Fitting decomposition theorem

Defines

Fitting’s decomposition theorem

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