  We show that any pseudo-effective divisor on a normal surface decomposes
uniquely into its "integral positive" part and "integral negative" part, which
is an integral analog of Zariski decompositions. By using this decomposition,
we give three applications: a vanishing theorem of divisors on surfaces (a
generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type
theorems of adjoint linear systems on surfaces (including a log version and a
relative version of the original one) and extension theorems of morphisms
defined on curves on surfaces (generalizations of Serrano and Paoletti's
results).
