Would there be homomorphism?

Question

$H=\left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}:a,b,c\in Z\right\}$ get the ring $\varphi :H\rightarrow Z$ , $\varphi \left( \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}\right) =c$ Does the transformation become a ring homomorphism? Essays: $\varphi \left( \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}+\begin{bmatrix} d & e \\ 0 & f \end{bmatrix}\right)=\varphi \left( \begin{bmatrix} a & b \\ o & c \end{bmatrix}\right) +\varphi \left( \begin{bmatrix} d & e \\ 0 & f \end{bmatrix}\right)$ because $c\neq 2c$ there is no homomorphism with interest, right? Is there something I can't see?