On bicommutators of modules over \(H\)-separable extension rings
type: Item
instance of: scholarly article
title: On bicommutators of modules over \(H\)-separable extension rings
zbMATH Open document ID: 0749.16018
author: Kozo Sugano
published in: Hokkaido Mathematical Journal
publication date: 1992-06-28T00:00:00Z
review text: The main result of this paper is as follows. Let \(A\), \(B\) be rings such that \(A\) is an \(H\)-separable extension of \(B\) and is finitely generated projective as a left \(B\)-module. Let \(M\) be a left \(A\)-module, let \(A^*\) denote the bicommutator \(\text{Bic}(_ AM)=\text{End}(_ \Delta M)\), \(\Delta=\text{End}(_ AM)\), \(B^*=\text{Bic}(_ BM)\). Then \(A^*\) is an \(H\)-separable extension of \(B^*\), \(A^*\) is finitely generated projective over \(B^*\) and \(B^*=V_{A^*}(V_{A^*}(B^*))\) where \(V_{A^*}(S)\) is the centralizer of \(S\) in \(A^*\) for any \(S\subseteq A^*\). If, as well, \(B\) is a left \(B\)-direct summand of \(A\), then \(A=A^*\) if and only if \(B=B^*\).
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 38344
zbMATH Keywords: \(H\)-separable extension
zbMATH Keywords: bicommutator
zbMATH Keywords: finitely generated projective
zbMATH Keywords: centralizer
zbMATH Keywords: direct summand
reviewed by: Margaret Beattie
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381413983
OpenAlex ID: W4249645934
DOI: 10.14492/HOKMJ/1381413983
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description: scientific article; zbMATH DE number 38344