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In the branch of [[abstract algebra]] called [[ring theory]], the '''Akizuki–Hopkins–Levitzki theorem''' connects the [[descending chain condition]] and [[ascending chain condition]] in [[Module (mathematics)|modules]] over semiprimary rings. A ring ''R'' (with 1) is called '''semiprimary''' if ''R''/''J''(''R'') is [[semisimple algebra|semisimple]] and ''J''(''R'') is a [[nilpotent ideal]], where ''J''(''R'') denotes the [[Jacobson radical]]. The theorem states that if ''R'' is a semiprimary ring and ''M'' is an ''R'' module, the three module conditions [[noetherian module|Noetherian]], [[artinian module|Artinian]] and "has a [[composition series]]" are equivalent. Without the semiprimary condition, the only true implication is that if ''M'' has a composition series, then ''M'' is both Noetherian and Artinian.

The theorem takes its current form from a paper by Charles Hopkins and a paper by [[Jacob Levitzki]], both in 1939. For this reason it is often cited as the '''Hopkins–Levitzki theorem'''. However [[Yasuo Akizuki]] is sometimes included since he proved the result for [[commutative rings]] a few years earlier {{Harv|Lam|2001}}.

Since it is known that [[artinian ring|right Artinian rings]] are semiprimary, a direct corollary of the theorem is: a right Artinian ring is also [[noetherian ring|right Noetherian]]. The analogous statement for left Artinian rings holds as well. This is not true in general for Artinian modules, because there are [[Artinian module#Relation to the Noetherian condition|examples of Artinian modules which are not Noetherian]].

Another direct corollary is that if ''R'' is right Artinian, then ''R'' is left Artinian if and only if it is left Noetherian.

== Sketch of proof ==
Here is the proof of the following: Let ''R'' be a semiprimary ring and ''M'' left ''R''-module. If ''M'' is either Artinian or Noetherian, then ''M'' has composition series.<ref>{{harvnb|Cohn|2003|loc=Theorem 5.3.9}}</ref> (The converse of this is always true.)

Let ''J'' be the radical of ''R''. Set <math>F_i = J^{i-1}M/J^iM</math>. The ''R'' module <math>F_i</math> may then be viewed as an <math>R/J</math>-module because ''J'' is contained in the [[annihilator (ring theory)|annihilator]] of <math>F_i</math>. Each <math>F_i</math> is a semisimple <math>R/J</math>-module, because <math>R/J</math> is a semisimple ring. Furthermore since ''J'' is nilpotent, only finitely many of the <math>F_i</math> are nonzero. If ''M'' is Artinian (or Noetherian), then <math>F_i</math> has a finite composition series. Stacking the composition series from the <math>F_i</math> end to end, we obtain a composition series for ''M''.

==In Grothendieck categories==
There are a number of generalizations and extensions of the theorem. One concerns [[Grothendieck category|Grothendieck categories]]: If ''G'' is a Grothendieck category with an artinian generator, then every artinian object in ''G'' is noetherian.<ref>{{cite book|title=Ring and Module Theory|author=Toma Albu|editor=Toma Albu|publisher=Springer|year=2010|chapter=A Seventy Years Jubilee: The Hopkins-Levitzki Theorem|url=http://books.google.com/books?id=pwBF-FCLJ80C&lpg=PA7&dq=hopkins%20theorem%20grothendieck%20categories&pg=PA7#v=onepage&q=nastasescu&f=false}}</ref>

== See also ==
* [[Artinian module]]
* [[Noetherian module]]
* [[Composition series]]

==References==
{{reflist}}
*{{citation
|last=Cohn
|first =P.M.
|year=2003
|title=Basic Algebra: Groups, Rings and Fields
}}
* Charles Hopkins (1939) ''Rings with minimal condition for left ideals'', Ann. of Math. (2) 40, pages 712–730.
* [[T. Y. Lam]] (2001) ''A first course in noncommutative rings'', Springer-Verlag. page 55 ISBN 0-387-95183-0
* [[Jacob Levitzki|Jakob Levitzki]] (1939) ''On rings which satisfy the minimum condition for the right-hand ideals'', Compositio Math. 7, pages 214–222.

{{DEFAULTSORT:Hopkins-Levitzki theorem}}
[[Category:Theorems in abstract algebra]]
[[Category:Ring theory]]

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