= Solution
Let $T=\operatorname{Spec}R$ and $Z=\operatorname{Spec}(R/(r))$. The <idempotent> $e$ determines the disjoint <clopen sets> $Z_A=D_Z(e)$ and $Z_B=D_Z(1-e)$. Each is closed in $T$. Define the <open subsets>
$$
T_A=T\setminus Z_B,\qquad T_B=T\setminus Z_A.
$$
They cover $T$ and satisfy $T_A\cap T_B=T\setminus Z=D(r)$. The <global section> $r$ defines <morphisms of schemes> $T_A\to A$ and $T_B\to B$, both sending the coordinate $t$ to $r$. On their intersection $r$ is a <unit>, so both maps land in the common <open subscheme> $A\cap B=\operatorname{Spec}\mathbb Z[t,t^{-1}]$, where they agree. They therefore glue uniquely:
$$
\boxed{u_{r,e}:T\longrightarrow X.}
$$
For an entirely explicit description, choose any lift $a\in R$ of $e$. Then $a(1-a)\in(r)$ and
$$
T_A=D(r)\cup D(a),\qquad T_B=D(r)\cup D(1-a).
$$
The equality $a+(1-a)=1$ proves that these <open subsets> cover. The inclusion $D(a(1-a))\subseteq D(r)$ proves that their intersection is exactly $D(r)$. Their description as complements of $Z_A,Z_B$ proves independence of the chosen lift. In particular, no lift of $e$ to an <idempotent> of $R$ is required.
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