Ideals and quotients in crossed products of Hopf algebras (Q1204403)

This paper is concerned with the relationships between ideals of crossed products \(R\#_ \sigma H\) and ideals of \(R\), and with certain quotient rings of these algebras, where \(H\) is a Hopf algebra over a field \(k\) and \(R\) is an \(H\)-module algebra. The main Hopf algebras under concern here are of the form \(K\# kG\), where \(K\) is irreducible and \(G\) is a group. Pointed cocommutative Hopf algebras are of this form. In Theorem 2.5: Essentially outer actions of such Hopf algebras on prime algebras \(R\) lead to a particularly nice connection: every non-zero ideal of \(R\#_ \sigma H\) intersects \(R\) non-trivially. These conditions lead in turn to interesting connections between the quotient rings: (1) \(C(R\#_ \sigma H)=C(R)^ H\), where \(C\) denotes the extended center of the appropriate quotient ring. (2) \(Q(R\#_ \sigma H)=Q_ H(R)\#_ \sigma H\), under an additional condition on regular elements. More is said when \(H\) is irreducible and cocommutative and \(\sigma\) is trivial.