In abstract algebra, in particular ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings.
A ring R (with 1) is called semiprimary if R/J(R) is 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, 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 (a former doctoral student of George Abram Miller) and a paper by Jacob Levitzki, both in 1939.
However Yasuo Akizuki is sometimes included since he proved the result[1] for commutative rings a few years earlier, in 1935.
Since it is known that right Artinian rings are semiprimary, a direct corollary of the theorem is: a right Artinian ring is also right Noetherian.
The analogous statement for left Artinian rings holds as well.
Here is the proof of the following: Let R be a semiprimary ring and M a left R-module.
If M is either Artinian or Noetherian, then M has a composition series.
One concerns Grothendieck categories: if G is a Grothendieck category with an Artinian generator, then every Artinian object in G is Noetherian.