Exercise 2 — A Complete Introduction to Abstract Algebra

Let the sets

\displaystyle R

\displaystyle=\left\{\begin{pmatrix}x&y\\ 0&z\end{pmatrix}\mid x,y,z\in\mathbb{R}\right\}\text{ and}

\displaystyle I

\displaystyle=\left\{\begin{pmatrix}x&y\\ 0&0\end{pmatrix}\mid x,y\in\mathbb{R}\right\}.

It is given that $R$ is a subring of $\mathcal{M}_{2\times 2}(\mathbb{R})$ .

Show that $I$ is a subring of $R$ .
Show that $I$ is an ideal of $R$ .
Simplify
$\left(\begin{pmatrix}1&2\\ 0&1\end{pmatrix}+I\right)\left(\begin{pmatrix}1&-2\\ 0&1\end{pmatrix}+I\right)$
in $R/I$ .

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