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2013-11-19T17:04:53Z

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			In [[mathematics]], more specifically [[ring theory]] and the theory of [[nil ideal]]s, '''Levitzky's theorem''', named after [[Jacob Levitzki]], states that in a right [[Noetherian ring]], every nil one-sided ideal is necessarily [[Nilpotent ideal\|nilpotent]].<ref name="Ref_">{{harvnb\|Herstein\|1968\|loc=Theorem 1.4.5\|p=37}}</ref><ref name="Ref_a">{{harvnb\|Isaacs\|1993\|loc=Theorem 14.38\|p=210}}</ref> 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}}

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			Assume that ''R'' satisfies the [[ascending chain condition]] on [[annihilator (ring theory)\|annihilators]] of the form <math>\{r\in R\mid ar=0\}</math> where ''a'' is in ''R''. Then
			# Any nil one-sided ideal is contained in the lower nil radical Nil<sub>*</sub>(''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 {{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==
			*[[Nilpotent ideal]]
			*[[Köthe conjecture]]
			*[[Jacobson radical]]

			==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 <!-- received 1939 -->
			\| title = On multiplicative systems
			\| journal = Compositio Math.
			\| 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=[[American Mathematical Monthly\|The American Mathematical Monthly]] \| issn=0002-9890 \| volume=70 \| issue=3 \| pages=286 \| jstor=2313127 \| publisher=Mathematical Association of America}}

			{{DEFAULTSORT:Levitzky's Theorem}}
			[[Category:Ring theory]]
			[[Category:Theorems in abstract algebra]]

en>Brad7777: added Category:Types of functions using HotCat

2012-02-04T19:44:04Z

added Category:Types of functions using HotCat

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Kostant partition function - Revision history

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