Question

Question: (b) Prove that the set l of matrices of the form$\left(\begin{array}{cc}0& b\\ 0& 0\end{array}\right)$with$\mathit{b}\mathbf{\in }\mathbf{ℝ}$is an idealin the ring S.

Short answer

It is proved that the setlof matrices of the form $\left(\begin{array}{cc}0& b\\ 0& 0\end{array}\right)$with $b\in \mathrm{ℝ}$is an ideal in ringS.

Step-by-step solution

Determine theorem 6.2 is ideal

Note that,

$$\begin{gathered} \left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& a\\ 0& 0\end{array}\right] \\ \left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]=\left[\begin{array}{cc}0& c\\ 0& 0\end{array}\right] \end{gathered}$$

Such that,$\left(\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\right)\subseteq l$and if $\left[\begin{array}{cc}0& b\\ 0& 0\end{array}\right]\in l\subseteq S$, then, it would be

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

Therefore, $l=\left(\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\right)$

Hence, Theorem 6.2 is an ideal.