On bicommutators of modules over \(H\)-separable extension rings. III
type: Item
instance of: scholarly article
title: On bicommutators of modules over \(H\)-separable extension rings. III
zbMATH Open document ID: 0856.16011
author: Kozo Sugano
published in: Hokkaido Mathematical Journal
publication date: 1994-10-18T00:00:00Z
review text: This is the third paper of the author on the study of the properties of the bicommutator of modules [for part II cf. ibid. 20, No. 3, 601-608 (1991; Zbl 0749.16019)]. Here he improves some of the theorems in the first paper [ibid. 20, No. 3, 497-501 (1991; Zbl 0749.16018)] by looking at extensions that generalize the bicommutator extension situation. The author also studies the Galois condition of the extension of bicommutators of a left \(A\)-module \(M\). The idea is to translate the properties of the \(H\)-separable extension \(A\) over \(B\) to the corresponding extension \(A^*\) over \(B^*\), where \(A^*\) is the bicommutator of \(A\) in the \(A\)-module \(M\) (this is the endomorphism ring of \(M\) as a left module over the endomorphism ring of \(M\) as a left \(A\)-module), and \(B^*\) likewise. When the \(H\)-separable extension \(A\) over \(B\) has the extra condition that \(A\) is a left f.g. projective \(B\)-module, the extension \(A^*\) over \(B^*\) is also \(H\)-separable, as proved by the author in part I. This situation is assumed for the main results of this paper. First, the author shows that if \(B\) also satisfies the double centralizer condition in \(A\), then \(_AM\) has the double centralizer condition if and only if \(_BM\) has the double centralizer condition (or \(A\cong A^*\) iff \(B\cong B^*\)). As a result of the proof, the case when \(A\) is a right f.g. projective \(B\)-module also holds. Second, the author gives a one-to-one correspondence between the subrings of \(A^*\) containing \(A\) and the subrings of \(B^*\) containing \(B\) which commute with \(A\). This correspondence also preserves localizations. Third, the author shows that if the \(H\)-separable extension \(A\) over \(B\) is also a \(G\)-Galois extension, then the same occurs for the extension \(A^*\) over \(B^*\). The technique used is to prove the statements for some general extensions \(S\) over \(T\) and then apply them to the case \(S=A^*\) and \(T=B^*\). The general s…
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 640820
zbMATH Keywords: bicommutator of modules
zbMATH Keywords: extensions of bicommutators
zbMATH Keywords: endomorphism rings
zbMATH Keywords: \(H\)-separable extensions
zbMATH Keywords: double centralizer condition
zbMATH Keywords: Galois extensions
reviewed by: Ricardo Alfaro
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381412693
OpenAlex ID: W2027165768
DOI: 10.14492/HOKMJ/1381412693
description: scientific article; zbMATH DE number 640820