As a student learning scheme theory with a broader background in differential and complex geometry, I always find it useful to discover "constructions" of schemes which work just as well in the setting of locally ringed spaces (and so I can see how they work in manifolds). For example, the Zariski tangent space and open / closed immersions work just as well in general locally ringed spaces.
What are some other constructions / definitions that one would learn in a course of scheme theory that work as well in the more general setting of locally ringed spaces? I am also interested if you think that it would be useful to learn them in that context or if it would be detrimental. (Perhaps the notion you have in mind works beautifully in the context of schemes but doesn't work very well in ringed spaces.)