Ideal of Ring/Examples/Order 2 Matrices with some Zero Entries

Example of Ideal of Ring
Let $R$ be the set of all order $2$ square matrices of the form $\begin{pmatrix} x & y \\ 0 & z \end{pmatrix}$ with $x, y, z \in \R$.

Let $S$ be the set of all order $2$ square matrices of the form $\begin{pmatrix} x & y \\ 0 & 0 \end{pmatrix}$ with $x, y \in \R$.

Then $R$ is a ring and $S$ is an ideal of $R$.

Proof
Let $\begin{pmatrix} x_1 & y_1 \\ 0 & z_1 \end{pmatrix}, \begin{pmatrix} x_2 & y_2 \\ 0 & z_2 \end{pmatrix} \in R$.

Then:

Thus by the Subring Test $R$ is a subring of the ring of order $2$ matrices over $\R$.

We have that, for example, $\begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix} \in S$

Hence $S \ne \O$.

Let $\begin{pmatrix} x_1 & y_1 \\ 0 & z_1 \end{pmatrix}, \begin{pmatrix} x_2 & y_2 \\ 0 & z_2 \end{pmatrix} \in S$.

and so $S$ is closed under matrix subtraction.

Now let $\begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix} \in S$ for real $a, b$.

Let $\begin{pmatrix} x & y \\ 0 & z \end{pmatrix} \in R$.

Then we have:

and:

Thus, by the Test for Ideal, $S$ is an ideal of $R$.