Chapter 6.2, Problem 7E

Assume that the set $S=\left\{\left[\begin{array}{cc}x& y\\ 0& z\end{array}\right]|x,y,z\in \mathbb{Z}\right\}$ is a ring with respect to matrix addition and multiplication.

Verify that the mapping $\theta :S\to \mathbb{Z}$ defined by $\theta \left(\left[\begin{array}{cc}x& y\\ 0& z\end{array}\right]\right)=z$ is an epimorphism from $S$ to $\mathbb{Z}$.

Describe ker $\theta ,$ and exhibit an isomorphism from $S/\ker \theta$ to $\mathbb{Z}$.