Chapter 6.4, Problem 26E

$26$. a. Let $R=M_2(ℝ)$, and $M=\left\{\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\right\}$. Show that $M$ and $M_2(ℝ)$ are only ideals of

$M_2(ℝ)$ and hence $M$ is a maximal ideal.

b. Show that $R/M$ is not a field. Hence Theorem $6.22$ is not true if the condition that $R$ iscommutative is removed.

Theorem 6.22 Quotient Rings That are Fields.

Let $R$ be a commutative ring with unity, and let $M$ be an ideal of $R$. Then $R/M$ is a field if and only if $M$ is a maximal ideal of $R$.