Invariant subrings and Jacobson radicals of Noetherian Hopf module algebras.
The author studies classical ring theoretical properties of Noetherian \(H\)-module algebras, where \(H\) is a Hopf algebra over a commutative ring \(k\), that is finitely generated and projective when viewed as a \(k\)-module. In particular, several facts known for group actions and group graded rings are extended to the case of Noetherian \(H\)-module algebras. In particular, it is proved that if \(H\) is a semisimple Hopf algebra and \(I\) is an \(H\)-stable one sided ideal of a left (or right) Noetherian \(H\)-module algebra \(A\) such that \(I\cap A^H\) is nilpotent, then \(I\) is nilpotent. If, in addition, \(A\) is \(H\)-semiprime then the classical Artinian ring of quotients \(Q(A)\) of \(A\) is isomorphic with the right localization of \(A\) at the set \(\Sigma\subseteq A^H\) of all regular elements of \(A^H\). Moreover, the subalgebra \(Q(A)^H\) of \(Q(A)\) is the classical ring of quotients of \(A^H\). In case when \(H\) is a finite-dimensional co-semisimple Hopf algebra then the Jacobson radical \(J(A)\) of a left (or right) Noetherian \(H\)-module algebra \(A\) is stable under the action of \(H\).
scientific article; zbMATH DE number 6486902
scientific article