Levitzky's theorem
In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent.[1][2] Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution to one of Köthe's questions as described in (Levitzki 1945). The result was originally submitted in 1939 as (Levitzki 1950), and a particularly simple proof was given in (Utumi 1963).
Proof
See also
Notes
- ^ Herstein, Theorem 1.4.5, p. 37
- ^ Isaacs, Theorem 14.38, p. 210
References
- I. Martin Isaacs (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-534-19002-2
- I.N. Herstein (1968), Noncommutative rings (1st ed.), The Mathematical Association of America, ISBN 0-88385-015-X
- J. Levitzki (1950), "On multiplicative systems", Compositio Math. 8: 76–80, MR0033799, http://www.numdam.org/item?id=CM_1951__8__76_0.
- Levitzki, Jakob (1945), "Solution of a problem of G. Koethe", American Journal of Mathematics (The Johns Hopkins University Press) 67 (3): 437–442, doi:10.2307/2371958, ISSN 0002-9327, JSTOR 2371958, MR0012269
- Utumi, Yuzo (1963), "Mathematical Notes: A Theorem of Levitzki", The American Mathematical Monthly (Mathematical Association of America) 70 (3): 286, doi:10.2307/2313127, ISSN 0002-9890, JSTOR 2313127, MR1532056