The Stacks project

17.2 Pathology

A ringed space is a pair consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}$. We allow $\mathcal{O} = 0$ in the definition. In this case the category of modules has a single object (namely $0$). It is still an abelian category etc, but it is a little degenerate. Similarly the sheaf $\mathcal{O}$ may be zero over open subsets of $X$, etc.

This doesn't happen when considering locally ringed spaces (as we will do later).