Classical quotient rings of trivial extensions
type: Item
instance of: scholarly article
title: Classical quotient rings of trivial extensions
zbMATH Open document ID: 0602.16003
author: Kazunori Sakano
published in: Tsukuba Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \(M\) be an \(R\)-\(R\)-bimodule and let \(L\) be the trivial extension whose elements (with an obvious notation) are of the form \((r,m)\) with componentwise addition and \((r,m)(r',m') = (rr',rm'+mr')\). Let \(D\) be the set of regular elements \(c\) of \(R\) such that cm\(\neq 0\neq mc\) for every nonzero element \(m\) of \(M\). Suppose either that \(M\) is faithful or that \(M\) has finite Goldie dimension on both sides. It is shown that an element \((r,m)\) of \(L\) is regular if and only if \(r\) is in \(D\). Also \(L\) has a classical right quotient ring if and only if both \(R\) and \(M\) satisfy the right Ore condition with respect to \(D\), and the right quotient ring of \(L\) is itself a trivial extension in the natural way.
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 3970959
zbMATH Keywords: trivial extension
zbMATH Keywords: set of regular elements
zbMATH Keywords: Goldie dimension
zbMATH Keywords: classical right quotient ring
zbMATH Keywords: right Ore condition
reviewed by: A. W. Chatters
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160402
OpenAlex ID: W1609655021
DOI: 10.21099/TKBJM/1496160402
description: scientific article; zbMATH DE number 3970959