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.