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On a Result of Levitzki

Published online by Cambridge University Press:  20 November 2018

B. Felzenszwalb
B. Felzenszwalb*
Affiliation:
Instituto De Matemática, Universidade Federal do Rio de Janeiro, C.P. 1835 ZC-0020.000 Rio de Janeiro, R. J., Brazil
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Extract

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A well known result of Levitzki [2, Lemma 1.1] is the following:

Theorem. Let R be a ring and U a non-zero one-sided ideal of R. Suppose that given a∈U, an = 0 for a fixed integer n ≥ 1; then R has a non-zero nilpotent ideal.

The purpose of this note is to observe some additional results which are related to the above.

Theorem 1. Let R be a ring with no non-zero nil ideals and U an ideal of R. Suppose that a∊R is such that for every x∊U, axn(x) = 0 where n(x) ≥ 1 depends on x; then aU= Ua = 0.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 2 , 01 June 1978 , pp. 241 - 242
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Felzenszwalb, B., Rings Radical over Subrings, Israel J. Math. 23 (2) (1976) 156-164.Google Scholar
2. Herstein, I. N., Topics in Ring Theory, University of Chicago Press, 1969.Google Scholar