dense ring of linear transformations

dense ring of linear transformations

Let

D

be a

division ring

and

V

a

vector space

over

D

. Let

R

be a

subring

of the

ring of endomorphisms

(

linear transformations

)

End

D

⁡

(

V

)

of

V

. Then

R

is said to be a

dense ring of linear transformations

(over

D

) if we are given

1.

any positive integer

n

,

2.

any set

{

v

1

,

…

,

v

n

}

of

linearly independent

vectors in

V

, and

3.

any set

{

w

i

,

…

,

w

n

}

of arbitrary vectors in

V

,

then there exists an

element

f

∈

R

such that

f

⁢

(

v

i

)

=

w

i

for

⁢

i

=

1

,

…

,

n

.

Note that the linear independence of the

v

i

’s is essential in insuring the

existence

of a linear transformation

f

. Otherwise, suppose

0

=

∑

d

i

⁢

v

i

where

d

1

≠

0

. Pick

w

i

’s so that they are linearly independent. Then

0

=

f

⁢

(

∑

d

i

⁢

v

i

)

=

∑

d

i

⁢

f

⁢

(

v

i

)

=

∑

d

i

⁢

w

i

, contradicting the linear independence of the

w

i

’s.

The notion of “dense” comes from

topology

: if

V

is given the

discrete topology

and

End

D

⁡

(

V

)

the

compact-open topology

, then

R

is dense in

End

D

⁡

(

V

)

iff

R

is a dense ring of linear transformations of

V

.

Proof.

First, assume that

R

is a dense ring of linear transformations of

V

. Recall that the compact-open topology on

End

D

⁡

(

V

)

has

subbasis

of the form

B

⁢

(

K

,

U

)

:=

{

f

∣

f

⁢

(

K

)

⊆

U

}

, where

U

is open and

K

is

compact

in

V

. Since

V

is

discrete

,

K

is finite. Now, pick a point

g

∈

End

D

⁡

(

V

)

and let

B

=

⋃

α

∈

I

⋂

i

=

1

n

⁢

(

α

)

B

⁢

(

K

i

⁢

α

,

U

i

⁢

α

)

be a

neighborhood

of

g

,

I

some

index set

. Then for some

α

∈

I

,

g

∈

⋂

B

⁢

(

K

i

⁢

α

,

U

i

⁢

α

)

. This means that

g

⁢

(

K

i

⁢

α

)

⊆

U

i

⁢

α

for all

i

=

1

,

…

,

n

⁢

(

α

)

. Since each

K

i

⁢

α

is finite, so is

K

:=

⋃

K

i

⁢

α

. After some re-indexing, let

{

v

1

,

…

,

v

n

}

be a

maximal

linearly independent subset of

K

. Set

w

j

=

g

⁢

(

v

j

)

,

j

=

1

,

…

,

n

. By

assumption

, there is an

f

∈

R

such that

f

⁢

(

v

j

)

=

w

j

, for all

j

. For any

v

∈

K

,

v

is a

linear combination

of the

v

j

’s:

v

=

∑

d

j

⁢

v

j

,

d

j

∈

D

. Then

f

(

v

)

=

∑

d

j

f

(

v

j

)

=

∑

d

j

g

(

j

)

=

g

(

v

)

∈

U

i

⁢

α

for some

i

. This shows that

f

⁢

(

K

i

⁢

α

)

⊆

U

i

⁢

α

and we have

f

∈

⋂

B

⁢

(

K

i

⁢

α

,

U

i

⁢

α

)

⊆

B

.

Conversely

, assume that the ring

R

is a

dense subset

of the space

End

D

⁡

(

V

)

. Let

v

1

,

…

,

v

n

be linearly independent, and

w

1

,

…

,

w

n

be arbitrary vectors in

V

. Let

W

be the

subspace

spanned by

the

v

i

’s. Because the

v

i

’s are linearly independent, there exists a linear transformation

g

such that

g

⁢

(

v

i

)

=

w

i

and

g

⁢

(

v

)

=

0

for

v

∉

W

. Let

K

i

=

{

v

i

}

and

U

i

=

{

w

i

}

. Then the

K

i

’s are compact and the

U

i

’s are open in the discrete space

V

. Clearly

g

∈

{

h

∣

h

⁢

(

v

i

)

=

w

i

}

=

B

⁢

(

K

i

,

U

i

)

for each

i

=

1

,

…

,

n

. So

g

lies in the neighborhood

B

=

∩

B

⁢

(

K

i

,

U

i

)

⊆

End

D

⁡

(

V

)

. Since

R

is dense in

End

D

⁡

(

V

)

, there is an

f

∈

R

∩

B

. This implies that

f

⁢

(

v

i

)

=

w

i

for all

i

.
∎

Remarks.

•

If

V

is finite dimensional over

D

, then any dense ring of linear transformations

R

=

End

D

⁡

(

V

)

. This can be easily observed by using the second half of the proof above. Take a basis

v

1

,

…

,

v

n

of

V

and any set of

n

vectors

w

1

,

…

,

w

n

in

V

. Let

g

be the linear transformation that maps

v

i

to

w

i

. The above proof shows that there is an

f

∈

R

such that

f

agrees with

g

on the basis elements. But then they must agree on all of

V

as a result, which is precisely the statement that

g

=

f

∈

R

.

•

It can be shown that a ring

R

is a

primitive ring

iff it is

isomorphic

to a dense ring of linear transformations of a vector space over a division ring. This is known as the

. It is a

generalization

of the special case of the

Wedderburn-Artin Theorem

when the ring in question is a simple Artinian ring. In the general case, the

finite chain

condition is dropped.

Title

dense ring of linear transformations

Canonical name

DenseRingOfLinearTransformations

Date of creation

2013-03-22 15:38:46

Last modified on

2013-03-22 15:38:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16K40

Synonym

dense ring

Related topic

SchursLemma

Defines

Jacobson Density Theorem

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