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Proceedings of the American Mathematical Society

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

 
 

 

Unimodular commutators


Author: Morris Newman
Journal: Proc. Amer. Math. Soc. 101 (1987), 605-609
MSC: Primary 15A36; Secondary 20H05
DOI: https://doi.org/10.1090/S0002-9939-1987-0911017-6
MathSciNet review: 911017
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Abstract: Let $ R$ be a principal ideal ring and $ {M_{k,n}}$ the set of $ k \times n$ matrices over $ R$. The following statments are proved: (a) If $ k \leq n/3$ then any primitive element of $ {M_{k,n}}$ occurs as the first $ k$ rows of the commutator of two elements of $ {\text{SL(}}n,R{\text{)}}$. (b) If every element of $ {\text{SL(}}3,R{\text{)}}$ is the product of at most $ {c_3}$ commutators, then every element of $ {\text{SL(}}n,R{\text{)}}$ is the product of at most $ {c_n}$ commutators, where $ {c_n} < c\log n + {c_3} - 3,c = 2\log (3/2) = 4.932 \ldots $, and $ n \geq 3$. (c) If $ n \geq 3$, then every element of $ {\text{SL(}}n,Z{\text{)}}$ is the product of at most $ c\log n + 40$ commutators, where $ c$ is given in (b) above


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DOI: https://doi.org/10.1090/S0002-9939-1987-0911017-6
Article copyright: © Copyright 1987 American Mathematical Society

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