The rings 2Z and 3Z are not isomorphic

The Mathematician — Thu, 09 Mar 2023 13:00:37 +0000

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.

2Z and 3Z are not isomorphic Archives - Epsilonify