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There's a exercise in my textbook, asking to show that $\mathbb{Z} [i] / \left\langle a + bi \right\rangle$ is not isomorphic to $\mathbb{Z}_n$ for any integer $n$ when $a$ and $b$ are not coprime. I've seen the related questions on this site, and I'm not seeing any help that is at the level of the textbook.

I'm able to establish that, when $a$ and $b$ are coprime, $\mathbb{Z} [i] / \left\langle a + bi \right\rangle \simeq \mathbb{Z}_{a^2 + b^2}$. Further, I believe that the idea is to use the properties that $\mathbb{Z} [i]$ has as a Euclidean domain, but I'm not sure how to move forward. Any hints or help are greatly appreciated.

Jane Doe
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If $a,b,d$ are nonzero integers, then you have the projection $$\mathbb{Z}[i]/\left\langle d(a + bi) \right\rangle\to \mathbb{Z}[i]/\left\langle d \right\rangle\cong \mathbb{Z}_d[i]$$

Now consider these rings as additive groups to see the the first quotient cannot be $\mathbb{Z}_n$ for some $n$, unless $d=\pm 1$.

Ofir
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