# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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 R. K. Sharma and J. B. Srivastava
Proc. Amer. Math. Soc. 94 (1985), 1-8 Request permission

## Abstract:

Let $\mathcal {L}(R)$ denote the associated Lie ring of an associative ring $R$ with identity $1 \ne 0$ under the Lie multiplication $[x,y] = xy - yx$ with $x,y \in R$. Further, suppose that the Lie ring $\mathcal {L}(R)$ is solvable of length $n$. It has been proved that if 3 is invertible in $R$, then the ideal $J$ of $R$ generated by all elements $[[[{x_1},{x_2}],[{x_3},{x_4}]],{x_5}],\;{x_1},\;{x_2},\;{x_3},\;{x_4},\;{x_5} \in R$, is nilpotent of index at most $\tfrac {2}{9}(19 \cdot {10^{n - 3}} - 1)$ for $n \geqslant 3$. Also, if 2 and 3 are both invertible in $R$, then the ideal $I$ of $R$ generated by all elements $[x,y],\;x,y \in R$, is a nil ideal of $R$. Some applications to Lie solvable group rings are also given.
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