A scheme with not-numerable affine covering.

Question

This question maybe finds the answer in the definition. Let $X$ be a scheme with not hypothesis on it (not noetherian, not affine,... ). Can I find a scheme that has a not numerable affine covering? For example, if a take a set of rings $\{R_i\}_{i \in I}$ such that $|I|=|\mathbb{R}|$, is it true that $\bigsqcup_{i \in I} \mathrm{Spec}(R_i)$ is a scheme?

Answer 1

Yes, $X=\bigsqcup \text{Spec } R_i$ is a scheme.