hollow matrix rings

hollow matrix rings

1

Definition

Definition 1

.

Suppose that

R

⊆

S

are both rings. The

hollow matrix ring

of

(

R

,

S

)

is the ring of matrices:

[

S

S

0

R

]

:=

{

[

s

t

0

r

]

:

s

,

t

∈

S

,

r

∈

R

}

.

It is easy to check that this forms a ring under the usual

matrix addition

and

multiplication

.
This definition is slightly simplified from the obvious higher dimensional examples
and the

transpose

of these matrices will also qualify as a hollow matrix ring.

The hollow matrix rings are highly counter-intuitive despite their simple definition.
In particular, they can be used to prove that in general a ring’s

left ideal

structure need not relate to its

right ideal

structure. We highlight a few
examples of this.

2

Left/Right Artinian and Noetherian

We specialize to an example with the fields

ℚ

and

ℝ

, though
the same argument can be made in much more general settings.

R

:=

[

ℝ

ℝ

0

ℚ

]

=

{

[

a

b

0

c

]

:

a

,

b

∈

ℝ

,

c

∈

ℚ

}

.

(1)

Claim 2

.

R

is

left Artinian

and

left Noetherian

.

Proof.

Let

I

be a left ideal of

R

and suppose
that

r

:=

[

x

y

0

z

]

∈

I

for some

x

,

y

∈

ℝ

and

z

∈

ℚ

.

Suppose that

z

≠

0

. Hence,

s

q

:=

[

0

0

0

q

/

z

]

∈

R

for each

q

∈

ℚ

and so

s

q

⁢

r

=

[

0

0

0

q

]

∈

I

for all

q

∈

ℚ

. In particular,

[

x

y

0

0

]

=

r

-

s

1

⁢

r

∈

I

. So in all cases it follows that

[

x

y

0

0

]

∈

I

. So now we take

r

=

[

x

y

0

0

]

and assume that

I

does not contain any

r

with

z

≠

0

.
By observing

matrix multiplication

it follows that

I

is now a left

ℝ

-

vector space

, and so any
chain of left

R

-modules is a chain of

subspaces

. As

dim

ℝ

⁡

I

≤

2

,
it follows that such chains are finite.

Hence, there can be no infinite descending chain of distinct left ideals
and so

R

is left Artinian and

Noetherian

.
∎

Claim 3

.

R

is not right Artinian nor right Noetherian.

Proof.

Using

π

(the usual

3.14

⁢

…

), or any other

transcendental number

, we define

I

n

:=

[

0

ℚ

⁢

[

π

;

n

]

0

0

]

,

(2)

where

ℚ

⁢

[

π

;

n

]

:=

{

q

⁢

(

π

)

⁢

π

n

:

q

⁢

(

x

)

∈

ℚ

⁢

[

x

]

}

.

(3)

Since

ℚ

⁢

[

π

;

n

]

properly contains

ℚ

⁢

[

π

;

n

+

1

]

for all

n

∈

ℤ

, it follows that

{

I

n

:

n

∈

ℤ

}

is an infinite proper
ascending and descending chain of right ideals. Therefore,

R

is neither right Artinian
nor right Noetherian.
∎

Corollary 4

.

R

does not have a ring

anti-isomorphism

. Thus

R

is not an

involutory ring

.

Proof.

If

R

is a ring with an anti-isomorphism, then the set of left ideals
is mapped to the set of right ideals, bijectively and order preserving.
This is not possible with

R

.
∎

Title

hollow matrix rings

Canonical name

HollowMatrixRings

Date of creation

2013-03-22 17:42:21

Last modified on

2013-03-22 17:42:21

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

10

Author

Algeboy (12884)

Entry type

Example

Classification

msc 16P20

Classification

msc 16W10

Related topic

InvolutaryRing

Related topic

Artinian

Related topic

Noetherian2

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