Yang, Shilin; Wang, D. Normal smash products. (English) Zbl 0910.16014 Port. Math. 55, No. 3, 323-331 (1998). Let \(H\) be a co-Frobenius Hopf algebra over a field (i.e. \(H\) has non-zero integrals) and \(A\) a right \(H\)-comodule algebra. The existence of a distinguished group-like element of \(H\) is proved, and this is used to construct a Morita context connecting the subalgebra \(A_1\) of coinvariants and the generalized smash product \(A\#H^{*\text{rat}}\). If \(\Phi\) is a normal class of prime rings, \(N\) a subgroup of group-like elements of \(H\), and \(A_N\) the \(N\)-graded algebra of \(N\)-coinvariants of \(A\), it is proved that \(A\#H^{*\text{rat}}\in\Phi\) if and only if \(A\) is faithful as both a left and a right \(A\#H^{*\text{rat}}\)-module and the graded smash product \(A_N\#N^*\in\Phi\). For a right \(H\)-Galois extension \(A/A_1\), the authors show that if \(A_1\) is central simple then so is \(A\#H^{*\text{rat}}\), and if \(A_1\) is a divisible ring, then \(A\#H^{*\text{rat}}\) is a dense ring of linear transformations of \(A\) over \(A_1\). Reviewer: Sorin Dascalescu (Bucureşti) Cited in 1 Review MSC: 16S40 Smash products of general Hopf actions 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W50 Graded rings and modules (associative rings and algebras) Keywords:co-Frobenius Hopf algebras; Morita contexts; normal classes of prime rings; central simple algebras; graded smash products; group-like elements; algebras of coinvariants; graded algebras PDFBibTeX XMLCite \textit{S. Yang} and \textit{D. Wang}, Port. Math. 55, No. 3, 323--331 (1998; Zbl 0910.16014) Full Text: EuDML