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

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:
 * $R / S \cong \R$

where:
 * $R / S$ is the quotient ring of $R$ by $S$
 * $\cong$ denotes ring isomorphism,

Proof
From Ideal of Ring: Order 2 Matrices with some Zero Entries:
 * $S$ is an ideal of $R$

Having defined the ring homomorphism $\phi: R \to \R$:
 * $\forall \mathbf A \in R: \map \phi {\begin {pmatrix} x & y \\ 0 & z \end {pmatrix} } = z$

from the First Isomorphism Theorem for Rings:
 * $\Img \phi \cong R / \map \ker \phi$

from which follows the result.