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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Automorphisms of $ {\rm GL}\sb{n}(R)$


Author: Bernard R. McDonald
Journal: Trans. Amer. Math. Soc. 215 (1976), 145-159
MSC: Primary 20G35
DOI: https://doi.org/10.1090/S0002-9947-1976-0382467-1
MathSciNet review: 0382467
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Abstract: Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair $ \langle R,S\rangle $ is said to be stable if the ring of fractions of R, $ {S^{ - 1}}R$, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair $ \langle R,S\rangle $ assume 2 is a unit in R and V is a free R-module of dimension $ \geqslant 3$. This paper examines the action of a group automorphism of $ GL(V)$ (the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of $ GL(V)$.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0382467-1
Keywords: Automorphisms, general linear group, commutative ring
Article copyright: © Copyright 1976 American Mathematical Society

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