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A locally ringed space is a ringed space whose stalk ${\mathcal{O}}_{X,x}$ is a local ring at every point. Morphisms of locally ringed spaces induce local homomorphisms on stalks.

For a morphism of ringed spaces, there is a natural morphismIt is an isomorphism when $\mathcal{E}$ is a locally free sheaf of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

For a ringed space $(X,{\mathcal{O}}_X)$, a sheaf of ${\mathcal{O}}_X$-modules is a sheaf $\mathcal{F}$ such that every $\mathcal{F}(U)$ is an ${\mathcal{O}}_X(U)$-module and restriction maps preserve scalar multiplication.

A sheaf of rings assigns a ring $\mathcal{O}(U)$ to every open set, restriction homomorphisms to inclusions, and satisfies local identity and gluing. It supplies the local functions on a ringed space.