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).

This is Utumi's argument as it appears in (Lam 2001, p. 164-165) Assume that R satisfies the ascending chain condition on annihilators of the form

Proof: In view of the previous lemma, it is sufficient to show that the lower nilradical of R is nilpotent.