# 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. 94 (1985), 39-45 Request permission

## Abstract:

Let $R$ be a principal ideal ring, ${M_{t,n}}$ the set of $t \times n$ matrices over $R$. The following results are proved: (a) Let $D \in {M_{n,n}}$. Then the least nonnegative integer $t$ such that a matrix $\left [ {\begin {array}{*{20}{c}} * & * \\ * & D \\ \end {array} } \right ]$ exists which belongs to ${\text {GL(}}n + t,R)$ is $t = n - p$, where $p$ is the number of invariant factors of $D$ equal to 1. (b) Any primitive element of ${M_{1,2n}}$ may be completed to a $2n \times 2n$ symplectic matrix. (c) If $A,B \in {M_{n,n}}$ are such that $[A,B]$ is primitive and $A{B^T}$ is symmetric, then $[A,B]$ may be completed to a $2n \times 2n$ symplectic matrix. (d) If $A \in {M_{t,t}},B \in {M_{t,n - t}}$, are such that $[A,B]$ is primitive and $A$ is symmetric, then $[A,B]$ may be completed to a symmetric element of ${\text {SL(}}n,R{\text {)}}$, provided that $1 \leqslant t \leqslant n/3$. (e) If $n \geqslant 3$, then any primitive element of ${M_{1,n}}$ occurs as the first row of the commutator of two elements of ${\text {SL(}}n,R{\text {)}}$.
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