Let $A$ be a Euclidean domain such that its only invertible elements are $1$ and $-1$, and let $\varphi : A^* \to \mathbb{N}$ be a Euclidean function. Show that if $a \in A^*$ is a non-invertible element for which $\varphi(a)$ is the smallest possible, then $A/\left<a\right>$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/3\mathbb{Z}$.
So far, I have proven that for all $b \in A$, if $b \notin <a>$, at least one of these cases happens: $(b-1) \in <a>$ or $(b+1) \in <a>$. I am trying to prove that these cases are mutual or exclusive, but I don't know how. Anyone can help me?
Let $x\in A$. Then $x=aq+r$ with $r=0$ or $\varphi(r)<\varphi(a)$. If $r\ne0$ then $r$ is invertible, so $r=\pm1$. It follows $|A/(a)|\le3$.