I think this question should already be abound in literature but the only place I find is from this article:

http://math.uchicago.edu/~lxiao/files/notes/Fundamental%20Groups.pdf

which seems to be elaborating this definition:

http://en.wikipedia.org/wiki/%C3%89tale_fundamental_group

but unfortunately as I do not understand much algebraic geometry I do not how to make use of this definition.

I am thinking about extending classical Bott periodicity to arbitrary rings that is good enough (UFD, for example). By extending I mean that I want to measure infinite matrices of entires in a ring $R$ with determinant 1 by the "one point compactification" of $R^{n}$ via introducing some topology. Hence in the classical case we can measure $U$ by $S^{n}$. I want to ask:

1): Is this possible? ( I thought about it over a bus trip but do not know how to establish universal bundles if the base ring is discrete, so I am stuck in here).

2): Is there any previous such constructions? What are their properties?

I feel there must be something well-known because Bott-periodicity theorem is a very old theorem. I do not know whether this is more appropriate for MO or for stack exchange, but I decided to put in here.