Summary: ``If an algebra $A$ is quantum commutative with respect to the action of a quasitriangular Hopf algebra $H$, then the monoidal structure on the category $_H{\scr M}$ of modules over $H$ induces a monoidal structure on the category $_{A\#H}{\scr M}$ of modules over the associated smash product $A\#H$. The condition under which the braiding structure of $_H{\scr M}$ induces a braiding structure on $_{A\#H}{\scr M}$ is further investigated. Dually, the notion of quantum cocommutativity is introduced, and a similar result in this dual situation is obtained.''