On trivial extensions which are quasi-Frobenius ones
type: Item
instance of: scholarly article
title: On trivial extensions which are quasi-Frobenius ones
zbMATH Open document ID: 0531.16009
author: Kazuhiko Hirata
published in: Tsukuba Journal of Mathematics
publication date: 1982-01-01T00:00:00Z
review text: In this short note it is proved that the trivial extension \(R\ltimes M\) of a ring R by a bimodule \(_ RM_ R\) is a left quasi-Frobenius extension of R in the sense of \textit{B. Müller} [Math. Z. 85, 345-368 (1964; Zbl 0203.044)] if and only if there exists an idempotent e in the center C of R such that the eC-module \(M^ R=\{m\in M| mr=rm\) for all \(r\in R\}\) is finitely generated and projective of rank 1, and \(M\cong M^ R\otimes_ CR\). This result generalizes a similar characterization by \textit{Y. Kitamura} [Arch. Math. 34, 111-113 (1980; Zbl 0508.16013)] for \(R\ltimes M\) to be a Frobenius extension of R.
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16U99
zbMATH DE Number: 3842073
zbMATH Keywords: trivial extension
zbMATH Keywords: left quasi-Frobenius extension
zbMATH Keywords: idempotent
zbMATH Keywords: center
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496159450
OpenAlex ID: W581227353
DOI: 10.21099/TKBJM/1496159450
description: scientific article; zbMATH DE number 3842073