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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Inner derivations of division rings and canonical Jordan form of triangular operators


Author: Dragomir Ž. Đoković
Journal: Proc. Amer. Math. Soc. 94 (1985), 383-386
MSC: Primary 16A39; Secondary 15A33
MathSciNet review: 787877
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Abstract: Let $ D$ be a division ring and $ k$ its center. We show that a generalized canonical Jordan form exists for triangularizable matrices $ A$ over $ D$ which are algebraic over $ k$, i.e, satisfy $ f(A) = 0$ for some nonzero polynomial $ f$ over $ k$. This canonical form is a direct sum of generalized Jordan blocks $ {J_m}(\alpha ,\beta )$. This block is an $ m$ by $ m$ matrix whose diagonal entries are equal to $ \alpha $, those on the first superdiagonal are equal to $ \beta $, and all other entries are equal to zero. If $ \alpha $ is separable over $ k$ then we can choose $ \beta = 1$, but in general this cannot be done.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0787877-7
Keywords: Annihilator, Jordan-Hölder series, generalized Jordan blocks, irreducible polynomial, bounded module, similarity
Article copyright: © Copyright 1985 American Mathematical Society