 Solution

Every restriction map of the structure sheaf of a scheme is a unital ring homomorphism. Thus $p\cdot 1=0$ in $\Gamma (X,{\mathcal{O}}_X)$ implies the same identity in $\Gamma (U,{\mathcal{O}}_X)$ for every open $U$. This proves (ii)$\Rightarrow$(i), so (i) and (ii) are equivalent.