So I am asked to prove or disprove whether the following are isomorphic to one another as rings:
- $\mathbb{Z}/4\mathbb{Z}$
- $2\mathbb{Z} /8\mathbb{Z}$
- $3\mathbb{Z}/12\mathbb{Z}$
- $\mathbb{R}/4\mathbb{R}$
Now, as far as $\mathbb{R}/4\mathbb{R}$ goes, I am sure that this is not isomorphic to any of the others since $4\mathbb{R}$ contains a unity element and thus the ideal is the entire $\mathbb{R}$.
I know that $\mathbb{Z}/4\mathbb{Z}:=\{0,1,2,3\} \cong \mathbb{Z}_4$, $2\mathbb{Z}/8\mathbb{Z} :=\{0,2,4,6\}$, $3\mathbb{Z}/12\mathbb{Z}:=\{0,3,6,9\} $. Now,because there are only two groups up to isomorphism of order 4, I assume showing that 2 or 3 is isomorphic to $\mathbb{Z}_2\times \mathbb{Z}_2$ is sufficient to show that it is not isomorphic, but I am unsure no how to proceed.
Since we have that $4\mathbb{R}$ contains a unity element, we have thus have $\mathbb{R}/4\mathbb{R}=\mathbb{R}/\mathbb{R}=\{0\}$. This is sufficient to say it cannnot be isomorphic to the other three.
Now, $\mathbb{Z}/4\mathbb{Z} \ncong 2\mathbb{Z}/8\mathbb{Z}$ since $\mathbb{Z}/4\mathbb{Z} \cong \mathbb{Z}_4$ and thus has unity element $1$, meanwhile a $2\mathbb{Z}/8\mathbb{Z}$ does not have a unity element. Since isomorphism preserve structural properties, they cannot be isomorphic.
Similiarly, $3\mathbb{Z}/12\mathbb{Z} \ncong 2\mathbb{Z}/8\mathbb{Z}$ by the same argument as above, where $9$ is the unity element in $3\mathbb{Z}/12\mathbb{Z}$.
We then check if $\mathbb{Z}/4\mathbb{Z} \cong 3\mathbb{Z}/12\mathbb{Z}$.
$\phi: 0 \to 0$
$\phi: 1 \to 9$
$\phi: 2 \to 6$
$\phi: 3 \to 3$
Is clearly a bijection and checking the homomorphism property for rings holds, thus we have a ring isomorphism.
Thank you to everyone in the comments for the tips on how to proceed.