The group of automorphisms of certain subalgebras of matrix algebras
type: Item
instance of: scholarly article
title: The group of automorphisms of certain subalgebras of matrix algebras
zbMATH Open document ID: 0729.16024
DOI: 10.1016/0021-8693(91)90206-N
author: Søren Jøndrup
published in: Journal of Algebra
publication date: 1991-01-01T00:00:00Z
review text: Let A be a prime ring which is also a K algebra, where K is a commutative ring, and denote by \(T_ n(A)\) the ring of upper triangular matrices over A. The author shows that every automorphism of \(T_ n(A)\) can be obtained by first applying an automorphism to A and then an inner automorphism. He goes on to study automorphisms of \(T_ n(a,R)\), the ring of \(n\times n\) matrices over a ring R whose entries below the main diagonal lie in the ideal a of R. If R is a (commutative) unique factorization domain and p is a prime element of R, then every automorphism of \(T_ n((p),R)\) is composed by an automorphism of R fixing (p), an inner automorphism and an element of the automorphism class group of \(T_ n((p),R)\) which is cyclic of order n. When (p) is replaced by a prime ideal of height \(>1\), every R automorphism is inner. With the help of the Jung-van der Kulk theorem [cf. \textit{P. M. Cohn}, Free rings and their relations (2nd Ed. Academic Press 1985; Zbl 0659.16001), p. 348], this gives a complete description of the automorphisms of \(T_ n((x,y),K[x,y])\), where K is a field and x, y indeterminates.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 4204597
zbMATH Keywords: prime ring
zbMATH Keywords: ring of upper triangular matrices
zbMATH Keywords: inner automorphism
zbMATH Keywords: automorphisms
zbMATH Keywords: unique factorization domain
zbMATH Keywords: Jung-van der Kulk theorem
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
cites work: Q3807387
cites work: Automorphisms of algebras of upper triangular matrices
cites work: Q4108256
cites work: Q5639812
cites work: Separable algebras over commutative rings
cites work: Automorphisms of upper triangular matrix rings
cites work: Fixed rings of finite automorphism groups of associative rings
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description: scientific article; zbMATH DE number 4204597