# Tagged: ring isomorphism

## 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 $\Q$.

## Problem 170

Prove that the rings $2\Z$ and $3\Z$ are not isomorphic.