Injective sheaf of modules

On a ringed space $(X,{\mathcal{O}}_X)$, an injective module sheaf is an injective object in the category of sheaves of modules over ${\mathcal{O}}_X$. Its defining extension property concerns module-sheaf morphisms; it is distinct from injectivity solely in the category of sheaves of abelian groups.

For opens $U\subseteq V\subseteq X$, there is a monomorphism between the extensions by zero of ${\mathcal{O}}_U$ and ${\mathcal{O}}_V$. Morphisms from these sheaves into an injective sheaf of modules $I$ identify with sections on $U$ and $V$, respectively. The injective object extension property therefore makes the restriction $I(V)\to I(U)$ surjective, proving that $I$ is flasque. The argument applies to any ringed space.