The Stacks project

Example 29.7.4. Let $A$ be a ring and $X = \mathop{\mathrm{Spec}}(A)$. Let $f_1, \ldots , f_ n \in A$ and let $U = D(f_1) \cup \ldots \cup D(f_ n)$. Let $I = \mathop{\mathrm{Ker}}(A \to \prod A_{f_ i})$. Then the scheme theoretic closure of $U$ in $X$ is the closed subscheme $\mathop{\mathrm{Spec}}(A/I)$ of $X$. Note that $U \to X$ is quasi-compact. Hence by Lemma 29.7.3 we see $U$ is scheme theoretically dense in $X$ if and only if $I = 0$.