Question about closed subsets of affine schemes

Question

Let $R$ be a commutative ring and let $U = \operatorname{Spec} R[ \frac{X_0}{X_k}, \ldots, \frac{X_n}{X_k} ]$. Let $Z$ be a closed subset of $U$. I would like to know how I can show that $$ Z = \operatorname{Spec} R[ \frac{X_0}{X_k}, \ldots, \frac{X_n}{X_k} ]/J $$ where $$ J = I(Z). $$ Thank you.