matrix first isomorphism problem

Question

Let R be the set of all matrices of the form \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} over $\mathbb{Q}$, such that $\ a = d\ $ and $\ b = 0\ $. Let $\ I\ $ be the subset of $\ \mathbb{R}\ $, such that $\ a = d = 0 \ . $ Show that $$R/I \cong Q.$$ [HINT : Think about defining a homomorphism...]

Help with this please, I cannot come up with a mapping, I've been trying to use the first isomorphism theorem by coming up with a ring homomorphism of $\mathbb{R}$ to $\mathbb{Q}$ with my image being $\mathbb{Q}$ and kernel being $I$ but I can't come up with a mapping at all.

Answer 1

Consider the map $$\begin{pmatrix}a&0\\c&a\end{pmatrix} \mapsto a.$$

What is the kernel of the map?

Answer 2

Make sure you check $R$ is a ring and $I$ is an ideal of $R$ first. Then find a map $f$ from $R$ to $\mathbb{Q}$. Show that $f$ is a ring homomorphism, $f$ is surjective, and $\ker f=I$.