Question

(b) Show that $\mathit{M}\mathbf{\left(}\mathbf{ℝ}\mathbf{\right)}$ has no ideals except the zero ideal and $\mathit{M}\mathbf{\left(}\mathbf{ℝ}\mathbf{\right)}$ itself.

Short answer

$M\left(ℝ\right)$has no ideals except the zero ideal and $M\left(ℝ\right)$ itself.

Step-by-step solution

J contains a matrix A

Let J be a nonzero ideal and $A\in M\left(ℝ\right)$will have $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=\left[\begin{array}{cc}{a}_{21}& a_{22}\\ a_{11}& a_{12}\end{array}\right]$and role="math" localid="1654245018377" $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}{a}_{12}& a_{11}\\ a_{22}& a_{21}\end{array}\right]$

Thus, $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\in J$. Let $A\in J$such that $a_{11}=c\ne 0$ then

$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\left[\begin{array}{cc}{c}^{-1}& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\in J$

Therefore, $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\in J$.

Sum of the matrices

From step 1, it is clear that $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\in J$and their sum is,

$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\in J$

Thus, the multiplicative identity is contained in J and $J=M\left(ℝ\right)$.

Therefore, $M\left(ℝ\right)$ has no ideals except the zero ideal and $M\left(ℝ\right)$ itself.