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Proceedings of the American Mathematical Society

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

 

 

Lie solvable rings


Authors: R. K. Sharma and J. B. Srivastava
Journal: Proc. Amer. Math. Soc. 94 (1985), 1-8
MSC: Primary 16A68
MathSciNet review: 781044
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0781044-9
Keywords: Lie solvable ring, associated Lie ring
Article copyright: © Copyright 1985 American Mathematical Society