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. 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).

See also

• Nilpotent ideal
• Köthe conjecture
• Jacobson radical

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. MR 0033799. {{cite journal}}: External link in |title= (help)
• Levitzki, Jakob (1945), "Solution of a problem of G. Koethe", American Journal of Mathematics, 67 (3), The Johns Hopkins University Press: 437–442, doi:10.2307/2371958, ISSN 0002-9327, MR 0012269
• Utumi, Yuzo (1963), "Mathematical Notes: A Theorem of Levitzki", The American Mathematical Monthly, 70 (3), Mathematical Association of America: 286, doi:10.2307/2313127, ISSN 0002-9890, MR 1532056