Let $X$ be a scheme.
- if the underlying space of $X$ is the disjount union of spaces $S_1$ and $S_2$, is $X$ necessarily a disjoint union of schemes with underlying spaces $S_1$ and $S_2$?
- if the underlying space of $X$ is the direct product of spaces $S_1$ and $S_2$, is $X$ necessarily a direct product of schemes with underlying spaces $S_1$ and $S_2$?
- if the ring of global functions on $X$ is the direct product of commutative unital rings $R_1$ and $R_2$, is $X$ necessarily a disjoint union of schemes with rings of global functions $R_1$ and $R_2$?
- if the ring of global functions on $X$ is the tensor product (over $\mathbb{Z}$) of commutative unital rings $R_1$ and $R_2$, is $X$ necessarily a direct product of schemes with rings of global functions $R_1$ and $R_2$?
More generally, what about finite (co)limits?
I believe that the answer to the first question is yes; the ringed space $(S_1\sqcup S_2, \mathcal{O}_X|_{S_1}\oplus \mathcal{O}_X|_{S_2})$ is isomorphic to $X$ as a ringed space.