Let $T$ be the set up upper triangular real $2 \times 2$ matrices, and let $J$ be defined by
$$J = \left\{ \begin{bmatrix} 0 & k \\ 0 & 0 \end{bmatrix} : k \in \mathbb{R} \right\}$$
Prove that:
1) $J$ is an ideal of $T$
2) $T/J$ is isomorphic to $\mathbb{R} \times \mathbb{R}$ as a ring.
What I've Done
I've been able to show that T is a subring of all $2 \times 2$ matrices and I showed that J is an ideal of T. However, I'm stuck on showing that the factor ring $T/J$ is isomorphic to $\mathbb{R}^2$.
My current attempt
Let $N = T/J$ be the factor ring defined by
\begin{align} N & = \left\{ t + J: t \in T \right\} \\ & = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} + J : a, b, c \in \mathbb{R} \right\} \end{align}
with $ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} + J = \left\{ \begin{bmatrix} a & b + k \\ 0 & c \end{bmatrix} : \begin{bmatrix} 0 & k \\ 0 & 0 \end{bmatrix} \in J \right\}$. Let $\phi$ be a map from $N$ to $\mathbb{R}^2$. We show $\phi$ is an isomorphism. Let $a, b \in N$, then
$$ \phi(a \cdot b) = \phi \left( \begin{bmatrix} a_1 & a_2 + k_a \\ 0 & a_3 \end{bmatrix} \cdot \begin{bmatrix} b_1 & b_2 + k_b \\ 0 & b_3 \end{bmatrix} \right) $$
This is where I get stuck. I am attempting to show three things:
1) $\phi$ is a homomorphism
2) $ Ker(\phi) = \left\{0 \right\} $
3) $\phi$ maps $N$ onto $\mathbb{R}^2$.
I get stuck because I don't see how to show that $\phi$ satisfies the property $f(a \cdot b) = f(a) \cdot f(b)$ I feel like I have to pick a $\phi$ which i'm not sure how to do. Another argument I feel might be valid is that since $J$ is an ideal of $T$ this implies something about the kernel of $\phi$, but I'm not sure. Where am I going wrong?
Hint: Consider $\psi: T \to \mathbb{R} \times \mathbb{R}$ given by $$ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \mapsto (a,c) $$ Prove that $\psi$ is a surjective ring homomorphism with kernel $J$. Use the isomorphism theorems.
(Recall that multiplication in $\mathbb{R} \times \mathbb{R}$ is defined component-wise.)