# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by Morris Newman
Proc. Amer. Math. Soc. 101 (1987), 605-609 Request permission

## 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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