Generalized Smash Products

In this paper, we study the ring (D,B) and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of -rational left modules and the category of all (B,D)-Hopf modules BMD D. Cai and Chen have proved this result in the case B = D = A. Secondly they have proved that if A has a nonzero left integral then AA *rat is a dense subring of End k (A). We prove that (A,A) is a dense subring of End k (Q), where Q is a certain subspace of (A,A) under the condition that the antipode is bijective (see Theorem 18). This condition is weaker than the condition that A has a nonzero integral. It is well known the antipode is bijective in case A has a nonzero integral. Furthermore if A has nonzero left integral, Q can be chosen to be A (see Corollary 19) and (A,A) is both left and right primitive. Thus AA *rat (A,A) End k (A). Moreover we prove that the left singular ideal of the ring (A,A) is zero. A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional, namely the ring (A,A) has a finite uniform dimension.