Automorphism groups of generalized triangular matrix rings.
type: Item
instance of: scholarly article
title: Automorphism groups of generalized triangular matrix rings.
zbMATH Open document ID: 1222.16017
published in: Linear Algebra and its Applications
publication date: 2011-02-09T00:00:00Z
review text: A ring is called `strongly indecomposable' if it cannot be represented as a generalized triangular matrix ring \(\left(\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\right)\) for some rings \(R\) and \(S\) and a nonzero \(R\)-\(S\)-bimodule \(_RM_S\). Typical examples are endomorphism rings of vector spaces and semiprime indecomposable rings. The main result of the paper states that if \(R\) and \(S\) are strongly indecomposable rings, then the triangulation of the nontrivial generalized matrix ring \(\left(\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\right)\) is unique up to isomorphism. As a corollary, if \(A=\left(\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\right)\), with \(M\neq 0\) and \(R\) and \(S\) strongly indecomposable, then \(\Aut(A)\) is a subgroup of \(\Aut(R)\times\Aut(S)\times M\times\Aut(M)\), consisting of all quadruples \((\rho,\psi,m,\chi)\), where \(\rho\in\Aut(R)\), \(\psi\in\Aut(S)\), \(m\in M\) and \(\chi\in\Aut(M)\) such that \(\chi\) is an \(R\)-\(S\)-bimodule automorphism of \(M\) relative to \(\rho\) and \(\psi\).
reviewed by: Johan Meyer
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 16D20
zbMATH DE Number: 5848730
zbMATH Keywords: automorphism groups
zbMATH Keywords: triangular matrix rings
zbMATH Keywords: strongly indecomposable rings
zbMATH Keywords: bimodule isomorphisms
zbMATH Keywords: endomorphism rings
author: Ánh Pham Ngoc
author: L. van Wyk
MaRDI profile type: Publication
full work available at URL: j.laa.2010.10.007
OpenAlex ID: W1983657257
cites work: The isomorphism problem for incidence rings
cites work: Triangular matrix representations
cites work: The automorphism group of a structural matrix algebra
cites work: Do isomorphic structural matrix rings have isomorphic graphs?
cites work: Complete blocked triangular matrix rings over a Noetherian ring
cites work: Q4441347
cites work: Q4101984
cites work: Automorphisms of upper triangular matrix rings
cites work: Isomorphism of generalized triangular matrix-rings and recovery of tiles
cites work: A note on algebra automorphisms of triangular matrices over commutative rings
cites work: Artinian rings with restricted primeness conditions
DOI: 10.1016/J.LAA.2010.10.007
description: scientific article; zbMATH DE number 5848730