Lie solvable rings

Sharma, R. K.; Srivastava, J. B.

doi:10.1090/S0002-9939-1985-0781044-9

Lie solvable rings
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by R. K. Sharma and J. B. Srivastava

Proc. Amer. Math. Soc. 94 (1985), 1-8

DOI: https://doi.org/10.1090/S0002-9939-1985-0781044-9

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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.

References

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