The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.
Problem 525
Let
\[R=\left\{\, \begin{bmatrix}
a & b\\
0& a
\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\]
Then the usual matrix addition and multiplication make $R$ an ring.
Let
\[J=\left\{\, \begin{bmatrix}
0 & b\\
0& 0
\end{bmatrix} \quad \middle | \quad b \in \Q \,\right\}\]
be a subset of the ring $R$.
(a) Prove that the subset $J$ is an ideal of the ring $R$.
(b) Prove that the quotient ring $R/J$ is isomorphic to $\Q$.
Add to solve later 
