The idea of a locallyringed space is that of a rather general geometrical object. There is also some nontrivial general theory associated to them. One must be rather more careful in defining sheaf of local rings than with sheaf of groups or other algebraic structure. It is not at all the case that F(U) is a local ring for open sets U of X: a ring, yes.
One can trace this back to the logical form of the definition of local ring. In a clean logical formulation, it can be put as 'for all r in R, either there is s with sr = 1 or t with t(1r) = 1'. In topos theory it is shown how the theory therefore qualifies for a classifying topos, which parametrises local rings. The structure of a locally ringed space is equivalent to the right kind of morphism to this topos  which can also be identified via algebraic geometry.
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