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
