Question

$2\times 2$Is the set $J=\left\{\left(\begin{array}{cc}0& 0\\ 0& r\end{array}\right)\overline{)r\in \mathrm{ℝ}}\right\}$an ideal in the ring $M\left(\mathrm{ℝ}\right)$of matrices over $R$?

Short answer

It is proved $J$is not an ideal.

Step-by-step solution

Determine set is ideal in the ring or not

No, the set is not ideal in the ring.

For example, $\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\in J,\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)\in M_2\left(\mathrm{ℝ}\right)$, and

$\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)\notin J$

That means this is not for all $r\in M_2\left(\mathrm{ℝ}\right)$and $a\in J$.

Here,$ra\in J$and $ar\in J$. Therefore, $J$is not ideal.