Disproving a ring Homomorphism Problem My question is:
$\varphi=\left\{
\begin{array}{c l} 
     T  &\mbox{$\longrightarrow R$}  \\
     \begin{pmatrix} 
  a     & 0\\ 
  b & c 
\end{pmatrix} & \mbox{$\longrightarrow$  $\begin{pmatrix} 
  a     & b\\ 
  0 & c 
\end{pmatrix}$}
\end{array}\right.$
Show that this function is not a ring homomorphism
I get that the definition of $f$ a ring homomorphism is when satisfies
$f(a + b) = f(a) + f(b)$ for all $a$ and $b$ in $\mathbb{R}$ 
and $f(ab) = f(a) f(b)$ for all $a$ and $b$ in $\mathbb{R}$
But I can't see how to prove the original question. Any help will be appreciated.

I will call $f$ to your $\varphi$. Look that if you have:
$$
A= \begin{pmatrix} 
  1 & 0\\ 
  1 & 0 
\end{pmatrix},
B= \begin{pmatrix} 
  0 & 0\\ 
  1 & 1 
\end{pmatrix}
$$
Then,
$$\begin{pmatrix} 
  1 & 0\\ 
  1 & 0 
\end{pmatrix}*\begin{pmatrix} 
  0 & 0\\ 
  1 & 1 
\end{pmatrix}= \begin{pmatrix} 
  0 & 0\\ 
  0 & 0 
\end{pmatrix}$$
So 
$$f(A*B)=\begin{pmatrix} 
  0 & 0\\ 
  0 & 0 
\end{pmatrix}$$
But:
$$f(A)*f(B)=\begin{pmatrix} 
  1 & 1\\ 
  0 & 0 
\end{pmatrix}*\begin{pmatrix} 
  0 & 1\\ 
  0 & 1 
\end{pmatrix}= \begin{pmatrix} 
  0 & 2\\ 
  0 & 0 
\end{pmatrix}$$ 
Making a contradiction to supposing $f$ is an homomorphism.
