Originally Posted by

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

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

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

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

$\displaystyle \begin{pmatrix}0&0\\d&0\end{pmatrix}\begin{pmatrix }0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\0&d\end {pmatrix}$ -- 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