  In this paper which is the first of a series of papers on smooth structures,
the concepts of C-structures and smooth structures are introduced and studied.
The notion of smooth structure on semi-integral domains is given. It is shown
that each semi-integral domain which is not a field, admits a unique smooth
structure and a large class of non-polynomial smooth functions on some
semi-integral domains is constructed. A smooth function from Z-{0} into Z is
given which does not extend to a smooth function on Z. No concept from topology
is used. As an application, it is shown that: Theorem - Let M and N be finite
dimensional smooth manifolds. The algebra of real smooth functions on M (resp.
N) will be denoted by A (resp. B). Assume that T is a homomorphism from B into
A. Then, there exists exactly one smooth mapping f from M into N such that
T=f*.
