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Normal smash products. (English) Zbl 0910.16014

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 \(A_1\).

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)
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