Generalized smash products.
Doi introduced the definition of the generalized smash product \(\#(D,B)\): (1) It is equal to \(\Hom_k(D,B)\) as a vector space; (2) \((f\cdot g)(d)=\sum f(g(d_2)_1\rightharpoonup d_1)g(d_2)_0\) for any \(f,g\in\#(D,B)\), \(d\in D\), where \((B,\rho)\) is a right \(A\)-comodule algebra, and \((D,\rightharpoonup)\) is a left \(A\)-module coalgebra for a bialgebra \(A\). The author considers the ring \(\#(D,B)\). First, it is shown that the category of \(BU\)-modules is equivalent to both the category of \(\#\)-rational left modules and the category of all \((B,D)\)-Hopf modules \(_BM^D\). The case when \(D=B=A\) has been studied by Cai and Chen. Secondly it is shown \(\#(A,A)\) is a dense subring of \(\text{End}_k(Q)\), where \(Q\) is a certain subspace of \(\#(A,A)\) under the condition that the antipode is bijective. Furthermore if \(A\) has nonzero left integral, \(Q\) can be chosen to be \(A\) and \(\#(A,A)\) is both left and right primitive. In fact, Cai and Chen have already shown that if \(A\) has a nonzero left integral then \(A\#A^{*rat}\) is a dense subring of \(\text{End}_k(A)\). Moreover, the author gives that the left singular ideal of the ring \(\#(A,A)\) is zero. As a corollary, it is shown that the ring \(\#(A,A)\) has finite uniform dimension.
scientific article; zbMATH DE number 2119277