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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.{{harvnb|Herstein|1968|loc=Theorem 1.4.5|p=37}}{{harvnb|Isaacs|1993|loc=Theorem 14.38|p=210}} 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 {{harv|Levitzki|1945}}. The result was originally submitted in 1939 as {{harv|Levitzki|1950}}, and a particularly simple proof was given in {{harv|Utumi|1963}}.

Proof

This is Utumi's argument as it appears in {{harv|Lam|2001|loc=p. 164-165}}

;Lemma{{sfn|Lam|2001|loc=Lemma 10.29}}

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

  1. Any nil one-sided ideal is contained in the lower nil radical Nil*(R);
  2. Every nonzero nil right ideal contains a nonzero nilpotent right ideal.
  3. Every nonzero nil left ideal contains a nonzero nilpotent left ideal.

;Levitzki's Theorem {{sfn|Lam|2001|loc=Theorem 10.30}}

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

{{reflist|2}}

References

  • {{Citation

|first1=I. Martin|last1=Isaacs

| year = 1993

| title = Algebra, a graduate course

| edition = 1st

| publisher = Brooks/Cole Publishing Company

| isbn = 0-534-19002-2

}}

  • {{Citation

|first1=I.N.|last1=Herstein

| year = 1968

| title = Noncommutative rings

| edition = 1st

| publisher = The Mathematical Association of America

| isbn = 0-88385-015-X

}}

  • {{Citation|last1=Lam|first1=T.Y.| author1-link=Tsit Yuen Lam|year=2001|title=A First Course in Noncommutative Rings|publisher=Springer-Verlag|isbn=978-0-387-95183-6}}
  • {{Citation

|first1=J.|last1=Levitzki

| year = 1950

| title = On multiplicative systems

| journal = Compositio Mathematica

| volume = 8

| pages = 76–80

| mr = 0033799

| url=http://www.numdam.org/item?id=CM_1951__8__76_0

| postscript = .

}}

  • {{Citation | last1=Levitzki | first1=Jakob | author1-link=Jacob Levitzki | title=Solution of a problem of G. Koethe | doi=10.2307/2371958 | mr=0012269 | year=1945 | journal=American Journal of Mathematics | issn=0002-9327 | volume=67 | pages=437–442 | jstor=2371958 | issue=3 | publisher=The Johns Hopkins University Press}}
  • {{Citation | last1=Utumi | first1=Yuzo | title=Mathematical Notes: A Theorem of Levitzki | doi=10.2307/2313127 | mr=1532056 | year=1963 | journal=The American Mathematical Monthly | issn=0002-9890 | volume=70 | issue=3 | pages=286 | jstor=2313127 | publisher=Mathematical Association of America| hdl=10338.dmlcz/101274 | hdl-access=free }}

{{DEFAULTSORT:Levitzky's Theorem}}

Category:Theorems in ring theory