In the branch of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
called
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
, the Akizuki–Hopkins–Levitzki theorem connects the
descending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
and
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
in
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
over semiprimary rings. A ring ''R'' (with 1) is called semiprimary if ''R''/''J''(''R'') is
semisimple and ''J''(''R'') is a
nilpotent ideal In mathematics, more specifically ring theory, an ideal ''I'' of a ring ''R'' is said to be a nilpotent ideal if there exists a natural number ''k'' such that ''I'k'' = 0. By ''I'k'', it is meant the additive subgroup generated by the set of ...
, where ''J''(''R'') denotes the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definition ...
. The theorem states that if ''R'' is a semiprimary ring and ''M'' is an ''R'' module, the three module conditions
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
,
Artinian and "has a
composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natur ...
" 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
Jacob Levitzki, also known as Yaakov Levitsky ( he, יעקב לויצקי) (17 August 1904 - 25 February 1956) was an Israeli mathematician.
Biography
Levitzki was born in 1904 in the Russian Empire and emigrated to then Ottoman-ruled Palestine ...
, 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, 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. This is not true in general for Artinian modules, because there are
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'' a left ''R''-module. If ''M'' is either Artinian or Noetherian, then ''M'' has a composition series. (The converse of this is true over any ring.)
Let ''J'' be the radical of ''R''. Set
. The ''R'' module
may then be viewed as an
-module because ''J'' is contained in the
annihilator of
. Each
is a
semisimple -module, because
is a semisimple ring. Furthermore, since ''J'' is nilpotent, only finitely many of the
are nonzero. If ''M'' is Artinian (or Noetherian), then
has a finite composition series. Stacking the composition series from the
end to end, we obtain a composition series for ''M''.
In Grothendieck categories
Several generalizations and extensions of the theorem exist. One concerns
Grothendieck categories: if ''G'' is a Grothendieck category with an Artinian generator, then every Artinian object in ''G'' is Noetherian.
See also
*
Artinian module
*
Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the prop ...
*
Composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natur ...
References
*
* 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
*
Jakob Levitzki (1939) ''On rings which satisfy the minimum condition for the right-hand ideals'', Compositio Mathematica, v. 7, pp. 214222.
{{DEFAULTSORT:Hopkins-Levitzki theorem
Theorems in ring theory