1. ## ideals and units

The question is
Show that if I in M2(R) contains ( 1 0 ), then I=M2(R)
0 0

I know the fact that if I contains some unit then it would be equal to the ring R.

So,
( a b ) (1 0 ) = (a 0 )
c d 0 0 c 0

(1 0 ) ( A B ) = ( A B
0 0 C D 0 0

But how can i say that this is an ideal?

2. Originally Posted by dreamon
The question is
Show that if I in M2(R) contains ( 1 0 ), then I=M2(R)
0 0

I know the fact that if I contains some unit then it would be equal to the ring R.

So,
( a b ) (1 0 ) = (a 0 )
c d 0 0 c 0

(1 0 ) ( A B ) = ( A B
0 0 C D 0 0

But how can i say that this is an ideal?

$\forall\,a,b,c,d\in\mathbb{R}$ :

$\begin{pmatrix}a&0\\0&0\end{pmatrix}\begin{pmatrix }1&0\\0&0\end{pmatrix}=\begin{pmatrix}a&0\\0&0\end {pmatrix}$

$\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix }0&b\\0&0\end{pmatrix}=\begin{pmatrix}0&b\\0&0\end {pmatrix}$

$\begin{pmatrix}0&0\\c&0\end{pmatrix}\begin{pmatrix }1&0\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\c&0\end {pmatrix}$

$\begin{pmatrix}0&0\\d&0\end{pmatrix}\begin{pmatrix }0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\0&d\end {pmatrix}$ -- the right matrix here is in the ideal because of the 2nd equality above .

Sum all the above up and you'll get that any matrix is in the ideal so...

Tonio

3. Originally Posted by tonio
$\forall\,a,b,c,d\in\mathbb{R}$ :

$\begin{pmatrix}a&0\\0&0\end{pmatrix}\begin{pmatrix }1&0\\0&0\end{pmatrix}=\begin{pmatrix}a&0\\0&0\end {pmatrix}$

$\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix }0&b\\0&0\end{pmatrix}=\begin{pmatrix}0&b\\0&0\end {pmatrix}$

$\begin{pmatrix}0&0\\c&0\end{pmatrix}\begin{pmatrix }1&0\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\c&0\end {pmatrix}$

$\begin{pmatrix}0&0\\d&0\end{pmatrix}\begin{pmatrix }0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\0&d\end {pmatrix}$ -- the right matrix here is in the ideal because of the 2nd equality above .

Sum all the above up and you'll get that any matrix is in the ideal so...

Tonio
Dont i have to find
1 0
0 1
as the unit matrix ?

4. Originally Posted by dreamon
Dont i have to find
1 0
0 1
as the unit matrix ?

I don't know what for, but if you insist then just input $a=d=1\,,\,b=c=0$ in what I posted and we're done.

Tonio