## 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 $\Q$.