  The total coordinate ring TC(X) of a normal variety is a generalization of
the ring introduced and studied by Cox in connection with a toric variety.
Consider a normal projective variety X with divisor class group Cl(X), and let
us assume that it is a finitely generated free abelian group. We define the
total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the
sum as above is taken over all Weil divisors of X contained in a fixed complete
system of representatives of Cl(X). We prove that for any normal projective
variety X, TC(X) is a UFD, this is a corollary of a more general theorem that
is proved in the paper. (Berchtold and Haussen proved the unique factorization
for a smooth variety independently.) We also prove that for X, the blow up of
P^2 along a finite number of collinear points, TC(X) is Noetherian. We also
give an example that TC(X) is not Noetherian but oplus_n H^0 (X, O(nD)) is
Noetherian for any Weil divisor D.
