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).

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

Lemma
Assume that *R* satisfies the ascending chain condition on annihilators of the form
{
r
∈
R
∣
a
r
=
0
}
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
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.