How to prove $1$ is in a sum of two ideals related to two disjoint closed subsets of an affine scheme

Question

Let $R$ be a commutative ring and let $U = \operatorname{Spec} R[ \frac{X_0}{X_k}, \ldots, \frac{X_n}{X_k} ]$. Suppose $Z \cap U$ and $V( ( \frac{\ell_0}{X_k}, \ldots, \frac{\ell_r}{X_k} ) )$ are closed disjoint subsets of $U$. I would like to know how one can show $$ 1 \in I(Z \cap U) + ( \frac{\ell_0}{X_k}, \ldots, \frac{\ell_r}{X_k} ). $$ In the book there is no further assumption on $R$. Thank you.

Answer 1

The scheme-theoretic intersection of two closed subschemes $V(I)$ and $V(J)$ inside of some affine scheme $\operatorname{Spec} R$ is just $V(I+J)$. This is a definition. With this, can you see how to finish? Full solution under the spoiler text.

As the intersection of our two closed subschemes is empty, we know that the ideal of functions vanishing on their intersection is the whole ring. On the other hand, we know that this is the sum of the ideals of each component of the intersection - so that sum contains 1.