So.. I was thinking about something last night and literally have no idea where to start.
Can the ring axioms be connected to topology? If so, how?
Basically, the motivation for my question has to do with the fact that manifolds can be classified in different ways. Likewise, rings (specifically, finite rings) can also be classified in different ways.
Is there a place in mathematics that connects the idea of manifolds from topology to rings from abstract algebra?
Ah ha... thanks to Googling of scheme theory, I stumbled onto a good starting point:
http://en.wikipedia.org/wiki/Locally_ringed_space
Ringed spaces is what I think I'm trying to understand.