# 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

- Any nil one-sided ideal is contained in the lower nil radical Nil
_{*}(*R*); - Every nonzero nil right ideal contains a nonzero nilpotent right ideal.
- 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

- ↑ Herstein 1968, p. 37, Theorem 1.4.5
- ↑ Isaacs 1993, p. 210, Theorem 14.38
- ↑ Lam 2001, Lemma 10.29.
- ↑ 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, ISSN 0002-9890, JSTOR 2313127, MR 1532056