Abstract: It is proved that the category of the left comodules over the ×_R-bialgebra L is equivalent to the category of the left L^*-modules, and the monoidal functor from ^L_AM to A#L^*-module is equivalent.For the ×_R-Hopf algebras the generalization of the fundamental structure theorem of Hopf modules is given by introducing the definition of antipode.

Key words: ×_R-Hopf algebra, smash product, antipode, fundamental structure theorem

摘要： ×_R-双代数L的左模范畴与左L^*-模范畴是等价的。左(L，A)-Hopf模范畴同构于左A#L^*- 模范畴。在引入了对极的概念之后并将基本结构定理推广到×_R-Hopf代数上的Hopf模上。

关键词: smash积, 对极, 基本结构定理, ×_R-Hopf代数