Invariant subrings and Jacobson radicals of Noetherian Hopf module algebras.
type: Item
instance of: scholarly article
title: Invariant subrings and Jacobson radicals of Noetherian Hopf module algebras.
zbMATH Open document ID: 1338.16038
author: Serge Skryabin
published in: Israel Journal of Mathematics
publication date: 2015-09-29T00:00:00Z
review text: 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 \(H\).
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16W22
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 6486902
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Noetherian \(H\)-module algebras
zbMATH Keywords: semiprime rings
zbMATH Keywords: Hopf algebra actions
zbMATH Keywords: smash product algebras
zbMATH Keywords: nilpotent ideals
zbMATH Keywords: Artinian rings of quotients
zbMATH Keywords: regular elements
zbMATH Keywords: rings of invariants
MaRDI profile type: Publication
full work available at URL: s11856-015-1165-9
OpenAlex ID: W1979400582
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DOI: 10.1007/S11856-015-1165-9
description: scientific article; zbMATH DE number 6486902