Connectedness from global regular functions

If a nonempty scheme $X$ has no nontrivial idempotent in $\Gamma (X,{\mathcal{O}}_X)$, then its underlying topological space is connected. Indeed, a decomposition into two nonempty clopen subsets would produce the global regular function equal to zero on one component and one on the other. In particular, $\Gamma (X,{\mathcal{O}}_X)$ being a field implies connectedness.