Lectures on Rings and Modules

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for all square matrices

then

E

C

B.

If, further, 5.

AO (,"'l (OI)E\I"

we call

Ii\

::::>AE""',

We shall

an ideal set, proper in case

generally denote ideal sets by greek upper case letters

IT,n

etc. and use latin script letters for preideal sets. We note some elementary consequences of the definitions: a)

If

IT

column lies in

is an ideal set, any matrix with a zero row or IT.

and this shows that

For if A

A = (A,O)

is not full.

say, then

A = Al (I 0)

3S

b) and any

Let C

be an ideal set.

IT

Then for any square

A,B

of the appropriate size, IT=-

For, given

A,B,C,

write

IT.

=

A

C = (c) C l

U,

the first row in each case, then

(1) 0

0

where

a,c

is

V(:,

where the determinantal sum is with respect to the first row. Now

(

0

=

(0

c) (A

0\

1

0

I)

thus the second matrix on the right of (1) is not full, and lies in

Hence

IT.

By elementary transformations we see that the same argument applies to the other rows of

C.

So we can make them all

IT,

if

A

of the same size,

AB

and

0

and the result follows. c) matrix

B

Let Then

In an ideal set

A

(A 0) I B

IE;

II

IE;

II,

becomes in turn and similarly

and let

B

IT

then for any square BA

IT.

be any matrix of the same size.

and by elementary transformations this (0

I

BA

IE;

-AB)

B

II.

Hence

AB

to

II,

36

Let

IT

be an ideal set and write

is the set of all

"1

clear that

Til

nxn

matrices in

is an ideal in

R.

as a "first approximation" to

IT =

L