Dubrovin valuation skew group rings
type: Item
instance of: scholarly article
title: Dubrovin valuation skew group rings
zbMATH Open document ID: 1022.16016
author: Zhong Yi
publication date: 2003-08-07T00:00:00Z
review text: In 1982, \textit{N.~I.~Dubrovin} [see Tr. Mosk. Mat. O.-va 45, 265-280 (1982; Zbl 0531.16011)] introduced a noncommutative valuation ring, now known as a Dubrovin valuation ring, and he showed that \(R\) is such a ring if and only if it is a semihereditary local order in a simple Artinian ring. Using this result, the paper under review determines when certain skew group rings are Dubrovin valuation rings. Let \(R\) be a commutative ring which is an order in an Artinian ring and let \(G\) be a finite group. Let \(t\colon G\to\Aut(R)\) be an action of \(G\) on \(R\) and let \(R_tG\) be the corresponding skew group ring. One interesting result here is that \(R_tG\) is a Dubrovin valuation ring if and only if the crossed products \(R_t^\alpha G\) are Dubrovin valuation rings for each \(t\)-cocycle \(\alpha\).
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 1885834
zbMATH Keywords: Dubrovin valuation rings
zbMATH Keywords: skew group rings
zbMATH Keywords: crossed products
zbMATH Keywords: semihereditary local orders
zbMATH Keywords: simple Artinian rings
reviewed by: D. S. Passman
MaRDI profile type: Publication
full work available at URL: s101140200172
OpenAlex ID: W2316254776
cites work: Fixed rings of finite automorphism groups of associative rings
cites work: Q3995022
cites work: Q3882615
cites work: Q3331359
cites work: NONCOMMUTATIVE VALUATION RINGS IN SIMPLE FINITE-DIMENSIONAL ALGEBRAS OVER A FIELD
cites work: Dubrovin valuation rings and Henselization
cites work: Global dimensions of crossed products
cites work: Dubrovin valuation properties of skew group rings and crossed products
cites work: Homological dimension of skew group rings and crossed products
cites work: Skew group rings and maximal orders
cites work: Orders with finite global dimension
DOI: 10.1007/S101140200172
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1885834