  We show in this paper that the Briancon-Skoda theorem holds for all ideals in
F-rational rings of positive prime characteristic, and also in rings with
rational singularities which are of finite type over a field of characteristic
0.
  Moreover, in Gorenstein F-rational rings of characteristic p we show that in
many cases the bound given in the Briancon-Skoda theorem may be improved by at
least one.
