[Aegis]

Well it depends on how a ring or sphere world is built. The only feasible way that I can see is that you build large gravity generators in a grid, and attract matter to them( asteroids, space dust and such). These fill the spaces between the grid lines, and provide most of the material for the worlds. They can then be mined.
Now, even then the idea is farstretched. I love RW's and SW's, but in reality getting the material for them is always going to be problematic. After all they are HUGE. (The present Versions actually allow each inhabitent to have at least a continent size area to themselves, even when they're fully populated. They could easily hold trillions of people. )
However, given that you have built one, however implausible the building process, I see no reason why you can't generate resources there. They are not just a very big ship after all. They are worlds. Massive worlds.

-Aegis