Multiplicative semigroup automorphisms of upper triangular matrices over rings
type: Item
instance of: scholarly article
title: Multiplicative semigroup automorphisms of upper triangular matrices over rings
zbMATH Open document ID: 0938.16029
DOI: 10.1016/S0024-3795(98)10026-5
author: Zhang Xian
author: Chongguang Cao
published in: Linear Algebra and its Applications
publication date: 2000-03-21T00:00:00Z
review text: Let \(R\) be a ring with \(1\) and \(C\) a central subring of \(R\). Let \(T_n(R)\) be the \(C\)-algebra of upper triangular \(n\times n\) matrices over \(R\). Several authors have characterized \(C\)-automorphisms of \(T_n(R)\). The authors generalize \textit{S.~Jøndrup}'s result [in Linear Algebra Appl. 221, 205-218 (1995; Zbl 0826.16034)]. The result obtained by Jøndrup says that if \(R\) is a semiprime ring or a ring in which all idempotents are central, then every \(C\)-automorphism of \(T_n(R)\) is the composite of an inner automorphism and an automorphism induced from a \(C\)-automorphism of \(R\). In this paper it is proved that if \(n\geqslant 2\) and \(R\) is a semiprime ring or a ring in which all idempotents are central, then \(f\) is a semigroup automorphism of \(T_n(R)\) if and only if there exist a nonsingular matrix \(P\) in \(T_n(R)\) and a ring automorphism \(\alpha\) of \(R\) such that \(f(A)=P^{-1}A^\alpha P\) for all \(A\) from \(T_n(R)\).
reviewed by: Viorel Mihai Gontineac
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S60
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 1354772
zbMATH Keywords: semigroup automorphisms
zbMATH Keywords: ring automorphisms
zbMATH Keywords: upper triangular matrices
zbMATH Keywords: semiprime rings
zbMATH Keywords: central idempotents
MaRDI profile type: Publication
cites work: Automorphisms of algebras of upper triangular matrices
cites work: Automorphism groups of certain algebras of triangular matrices
cites work: Automorphisms and derivations of upper triangular matrix rings
cites work: The group of automorphisms of certain subalgebras of matrix algebras
cites work: A note on algebra automorphisms of triangular matrices over commutative rings
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description: scientific article; zbMATH DE number 1354772