= Affine line with doubled origin
{title2=$\mathbb A^1\amalg_{D(t)}\mathbb A^1$}
Glue two copies of the <affine line> over a base along the <principal open subset> $D(t)$ by the identity. This <scheme> has two copies of the zero section and a canonical <morphism of schemes> to the original <affine line>. Over $\mathbb Z$, its <functor of points> assigns to a <commutative ring> $R$ the pairs $(r,e)$ with $e$ an <idempotent> in $R/(r)$. Indeed, $r$ gives the map to the <affine line>, while $e$ partitions $V(r)$ into the two <clopen sets> assigned to the zero sections. Complementing the opposite part gives two <open subsets> covering $\operatorname{Spec}R$, whose intersection is $D(r)$; the coordinate $r$ defines maps on them which glue. This also proves that no lifting of $e$ to an <idempotent> of $R$ is necessary.
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