= Norm-Euclidean integers of discriminant minus seven
The <ring of integers> $\mathbb Z[(1+\sqrt{-7})/2]$ is a <Euclidean domain> for $N(z)=z\overline z$. In its lattice choose the nearest row, whose imaginary-coordinate error is at most $\sqrt7/4$, then the nearest point on that row, whose real-coordinate error is at most $1/2$. Every complex number is thus within squared distance $11/16<1$ of a lattice point, giving Euclidean division. The suborder $\mathbb Z[\sqrt{-7}]$ has the nonprincipal <ideal> $(2,1+\sqrt{-7})$ and consequently is not a <Euclidean domain> for any Euclidean function.
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