matrix ring

matrix ring

0.1

Matrix Rings

A ring

R

is said to be a

matrix ring

if there is a ring

S

and a positive integer

n

such that

R

≅

M

n

⁢

(

S

)

,

the ring of

n

×

n

matrices with entries as elements of

S

. Usually, we simply identify

R

with

M

n

⁢

(

S

)

.

Generally, one is interested to find out if a given ring

R

is a matrix ring. By setting

n

=

1

, we see that every ring is trivially a matrix ring. Therefore, to exclude these trivial cases, we call a ring

R

a

trivial matrix ring

if there does not exist an

n

>

1

such that

R

≅

M

n

⁢

(

S

)

. Now the question becomes: is

R

a non-trivial matrix ring?

Actually, the requirement that

S

be a ring in the above definition is redundent. It is enough to define

S

to be simply a set with two

binary operations

+

and

⋅

. Fix a positive integer

n

≥

1

, define the set of formal

n

×

n

matrices

M

n

⁢

(

S

)

with

coefficients

in

S

. Addition and multiplication on

M

n

⁢

(

S

)

are defined as the usual

matrix addition

and multiplication, induced by

+

and

⋅

of

S

respectively. By abuse of notation, we use

+

and

⋅

to denote addition and multiplication on

M

n

⁢

(

S

)

. We have the following:

1.

If

M

n

⁢

(

S

)

with

+

is an

abelian group

, then so is

S

.

2.

If in addition,

M

n

⁢

(

S

)

with both

+

and

⋅

is a ring, then so is

S

.

3.

If

M

n

⁢

(

S

)

is unital (has a

multiplicative identity

), then so is

S

.

The first two assertions above are easily observed. To see how the last one roughly works, assume

E

is the multiplicative identity of

M

n

⁢

(

S

)

. Next define

U

⁢

(

a

,

i

,

j

)

to be the matrix whose

(

i

,

j

)

-cell is

a

∈

S

and

0

everywhere else. Using cell entries

e

s

⁢

t

from

E

, we solve the system of equations

U

⁢

(

e

s

⁢

t

,

i

,

j

)

⁢

E

=

U

⁢

(

e

s

⁢

t

,

i

,

j

)

=

E

⁢

U

⁢

(

e

s

⁢

t

,

i

,

j

)

to conclude that

E

takes the form of a

diagonal matrix

whose

diagonal

entries are all the same element

e

∈

S

. Furthermore, this

e

is an

idempotent

. From this, it is easy to derive that

e

is in fact a multiplicative identity of

S

(multiply an element of the form

U

⁢

(

a

,

1

,

1

)

, where

a

is an arbitrary element in

S

). The converse of all three assertions are clearly true too.

Remarks

.

•

It can be shown that if

R

is a unital ring having a finite doubly-indexed set

T

=

{

e

i

⁢

j

∣

1

≤

i

,

j

≤

n

}

such that

(a)

e

i

⁢

j

⁢

e

k

⁢

ℓ

=

δ

j

⁢

k

⁢

e

i

⁢

ℓ

where

δ

j

⁢

k

denotes the

Kronecker delta

, and

(b)

∑

e

i

⁢

j

=

1

,

then

R

is a matrix ring. In fact,

R

≅

M

n

⁢

(

S

)

, where

S

is the

centralizer

of

T

.

•

A unital matrix ring

R

=

M

n

⁢

(

S

)

is

isomorphic

to the

ring of endomorphisms

of the

free module

S

n

. If

S

has IBN, then

M

n

⁢

(

S

)

≅

M

m

⁢

(

S

)

implies that

n

=

m

. It can also be shown that

S

has IBN iff

R

does.

•

Any ring

S

is

Morita equivalent

to the matrix ring

M

n

⁢

(

S

)

for any positive integer

n

.

0.2

Matrix Groups

Suppose

R

=

M

n

⁢

(

S

)

is unital.

U

⁢

(

R

)

, the

group of units

of

R

, being isomorphic to the group of

automorphisms

of

S

n

, is called the

general linear group

of

S

n

. A

matrix group

is a

subgroup

of

U

⁢

(

R

)

for some matrix ring

R

. If

S

is a field, in particular, the field of real numbers or complex numbers, matrix groups are sometimes also called

classical groups

, as they were studied as far back as the 1800’s under the name groups of tranformations, before the formal

concept

of a group was introduced.

Title

matrix ring

Canonical name

MatrixRing

Date of creation

2013-03-22 15:54:18

Last modified on

2013-03-22 15:54:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16S50

Defines

matrix group

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