Is there a version of the weak Nullstellensatz valid in general locally ringed spaces? The statement I imagine is something like this:
Let $(X, \mathcal{O}_X)$ be a locally ringed space (possibly with some assumptions). Let $I$ be an ideal in $\Gamma(X, \mathcal{O}_X)$, and suppose for all $p \in X$, there is some $f \in I$ such that $f|_p$, i.e. in the image of $f$ in $\mathcal{O}_{X, p}$, is a unit. Then $I = \Gamma(X, \mathcal{O}_X)$.
One can show that this is true if $I$ is a principal ideal (if $f$ is invertible in all stalks, then we can find an inverse around each point and glue them together).