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Concept# Levitzky's theorem

Summary

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 . The result was originally submitted in 1939 as , and a particularly simple proof was given in .
Proof
This is Utumi's argument as it appears in
;Lemma
Assume that R satisfies the ascending chain condition on annihilators of the form {r\in R\mid ar=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

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