Homomorphisms,separable extensions,and Morita maps for weak module algebras

By using a trace one element, we give a sufficient and necessary condition for a weak module algebra A to be a projective left A#H-module, where A#H denotes the weak smash product. We also give some differentiated conditions for the weak smash product to be a separable extension on the weak module algebra A and get the weak structure theorem in the category of weak (H,A)-Hopf modules.