  We associate to every divisorial (e.g. smooth) variety $X$ with only constant
invertible global functions and finitely generated Picard group a
$Pic(X)$-graded homogeneous coordinate ring. This generalizes the usual
homogeneous coordinate ring of the projective space and constructions of Cox
and Kajiwara for smooth and divisorial toric varieties. We show that the
homogeneous coordinate ring defines in fact a fully faithful functor. For
normal complex varieties $X$ with only constant global functions, we even
obtain an equivalence of categories. Finally, the homogeneous coordinate ring
of a locally factorial complete irreducible variety with free finitely
generated Picard group turns out to be a Krull ring admitting unique
factorization.
