Hopf crossed products, rings of quotients, and prime ideals
type: Item
instance of: scholarly article
title: Hopf crossed products, rings of quotients, and prime ideals
zbMATH Open document ID: 0829.16029
published in: Advances in Mathematics
publication date: 1995-05-15T00:00:00Z
review text: This interesting article is concerned with crossed products \(A = R \#_\sigma H\) of a Hopf algebra \(H\) over an algebra \(R\). The authors introduce two new ``actions'' of \(H\) on \(R\) which are extremely useful in simplifying computations involving the cocycle \(\sigma\). Using this technique, the original \(H\)-action on \(R\) is extended to various quotient rings of \(R\) and corresponding quotient rings of \(A\) are described. These results are then applied to study prime ideals of \(A\). The main result in this direction is the following (Theorem 7.12): Let \(H\) be semisimple and either commutative or cocommutative and assume that \(R\) is \(H\)-prime. Then \(A\) is semiprime, with at most \(\dim H\) many minimal primes, and a prime of \(A\) is minimal precisely if it intersects \(R\) trivially. Replacing the above assumption on \(H\) by the hypothesis that the \(H\)- action on \(R\) is inner, in a suitable sense, an analogous but slightly weaker result is obtained (\(A\) need not be semiprime). Various sufficient conditions for \(A\) to be semiprime are derived which greatly extend what was previously known on this subject, but the general question, as to whether \(A\) is semiprime if \(R\) is \(H\)-semiprime and \(H\) is semisimple, remains open.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 755386
zbMATH Keywords: actions of Hopf algebras
zbMATH Keywords: semiprime rings
zbMATH Keywords: crossed products
zbMATH Keywords: Hopf algebra
zbMATH Keywords: quotient rings
zbMATH Keywords: prime ideals
zbMATH Keywords: minimal primes
author: Susan Montgomery
author: Hans-Jürgen Schneider
reviewed by: Martin Lorenz
MaRDI profile type: Publication
full work available at URL: aima.1995.1027
OpenAlex ID: W1982972869
DOI: 10.1006/AIMA.1995.1027
description: scientific article; zbMATH DE number 755386