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. 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 $\$ 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.

See also

* Nilpotent ideal
* Köthe conjecture
* Jacobson radical

Notes

References

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{{DEFAULTSORT:Levitzky's Theorem
Theorems in ring theory