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Automorphisms of algebras of upper triangular matrices. (English) Zbl 0674.16014

In an earlier work Jøndrup considered the ring of upper triangular matrices with entries from a simple artinian ring which is finite dimensional over its center. He shows that every automorphism is inner. In this note we obtain a somewhat more general result. Let A be an algebra over a commutative ring R for which every nontrivial R- endomorphism of A is an R-automorphism. Then every R-automorphism of the algebra of upper triangular matrices can be factored into a product of an inner automorphism and an automorphism which is induced elementwise by an R-automorphism of A.
Reviewer: G.P.Barker

MSC:

16S50 Endomorphism rings; matrix rings
16W20 Automorphisms and endomorphisms
16P10 Finite rings and finite-dimensional associative algebras
15A30 Algebraic systems of matrices
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References:

[1] G. P.Barker and T. P.Kezlan, The automorphism group of a matrix algebra. In: Current Trends in Matrix Theory. Edited by F. Uhlig and R. Grone, New York 1987. · Zbl 0658.16027
[2] I. M. Isaacs, Automorphisms of matrix algebras over commutative rings. Linear Algebra Appl.31, 215-231 (1980). · Zbl 0434.16015 · doi:10.1016/0024-3795(80)90221-9
[3] I. M. Isaacs, Review z# 88h: 16029 (review of [1]), Math. Reviews88h, 4011 (1988).
[4] S. J?ndrup, Automorphisms of upper triangular matrix rings. Arch. Math.49, 497-502 (1987). · Zbl 0601.16028 · doi:10.1007/BF01194296
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