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

This is Utumi's argument as it appears in (Lam 2001, p. 164-165)

Lemma[3]

Assume that R satisfies the ascending chain condition on annihilators of the form where a is in R. Then

  1. Any nil one-sided ideal is contained in the lower nil radical Nil*(R);
  2. Every nonzero nil right ideal contains a nonzero nilpotent right ideal.
  3. Every nonzero nil left ideal contains a nonzero nilpotent left ideal.
Levitzki's Theorem [4]

Let R be a right Noetherian ring. Then every nil one-sided ideal of R is nilpotent. In this case, the upper and lower nilradicals are equal, and moreover this ideal is the largest nilpotent ideal among nilpotent right ideals and among nilpotent left ideals.

Proof: In view of the previous lemma, it is sufficient to show that the lower nilradical of R is nilpotent. Because R is right Noetherian, a maximal nilpotent ideal N exists. By maximality of N, the quotient ring R/N has no nonzero nilpotent ideals, so R/N is a semiprime ring. As a result, N contains the lower nilradical of R. Since the lower nilradical contains all nilpotent ideals, it also contains N, and so N is equal to the lower nilradical. Q.E.D.

See also

Notes
  1. Herstein 1968, p. 37, Theorem 1.4.5
  2. Isaacs 1993, p. 210, Theorem 14.38
  3. Lam 2001, Lemma 10.29.
  4. Lam 2001, Theorem 10.30.

References
  • Isaacs, I. Martin (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-534-19002-2
  • Herstein, I.N. (1968), Noncommutative rings (1st ed.), The Mathematical Association of America, ISBN 0-88385-015-X
  • Lam, T.Y. (2001), A First Course in Noncommutative Rings, Springer-Verlag, ISBN 978-0-387-95183-6
  • Levitzki, J. (1950), "On multiplicative systems", Compositio Mathematica, 8: 76–80, MR 0033799.
  • 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, MR 0012269
  • Utumi, Yuzo (1963), "Mathematical Notes: A Theorem of Levitzki", The American Mathematical Monthly, Mathematical Association of America, 70 (3): 286, doi:10.2307/2313127, hdl:10338.dmlcz/101274, ISSN 0002-9890, JSTOR 2313127, MR 1532056
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