Dual number
= Dual number
{title2=$k[\varepsilon]/(\varepsilon^2)$}
= Dual-number algebra
{synonym}
The dual-number <ring> over a <field> $k$ consists of elements $a+b\varepsilon$ with $\varepsilon^2=0$. The element $\varepsilon$ is a nonzero <nilpotent element>; its <ideal> is the <ring>'s only <prime ideal>. Hence its <affine scheme> has the same one-point topology as $\operatorname{Spec}k$ while its <structure sheaf> differs.