On finite morphisms of ruled surfaces

We prove that a surjective morphism Phi:X --> X between ruled surfaces is finite and it descends to a finite morphism phi:C'--> C between base curves of X' and X. When Phi is restricted to the fibres of X', it has a constant degree, say a, and then deg Phi=a deg phi. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C' and C are elliptic and X' is the fibre product C'x(C)X especially.