Prove that the rings 2\mathbb{Z} and 3\mathbb{Z} are not isomorphic.
Proof. Take the assumption that
2\mathbb{Z} and
3\mathbb{Z} are isomorphic, and take an arbitrary isomorphic mapping:
\begin{align*}
\phi: 2\mathbb{Z} \longrightarrow 3\mathbb{Z}.
\end{align*}
Now let
\phi(2) = 3a for some
a \in \mathbb{Z}. Then
\begin{align*}
\phi(4) = \phi(2 + 2) = \phi(2) + \phi(2) = 6a,
\end{align*}
where
\phi(2 + 2) = \phi(2) + \phi(2) follows by the ring homomorphism property, and
\begin{align*}
\phi(4) = \phi(2\cdot 2) = \phi(2)\phi(2) = 9a^2,
\end{align*}
where
\phi(2\cdot 2) = \phi(2)\phi(2) follows again by the ring homomorphism property. Then we see that
\begin{align*}
9a^2 = 6a \iff a = 0.
\end{align*}
This would mean that
\phi(0) = \phi(2) = 0, which contradicts the injectivity property of
\phi.
Therefore, the rings
2\mathbb{Z} and
3\mathbb{Z} are not isomorphic.
