Hopkins–Levitzki theorem
Encyclopedia
In the branch of abstract algebra
called ring theory
, the Akizuki-Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition
in modules
over semiprimary rings. A ring R 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
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 .
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.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
called ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, the Akizuki-Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition
Ascending chain condition
The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
in modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over semiprimary rings. A ring R is called semiprimary if R/J(R) is semisimple
Semisimple algebra
In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...
and J(R) is a nilpotent ideal
Nilpotent ideal
In mathematics, more specifically ring theory, an ideal, I, of a ring is said to be a nilpotent ideal, if there exists a natural number k such that Ik = 0. By Ik, it is meant the additive subgroup generated by the set of all products of k elements in I...
, where J(R) denotes the Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
. The theorem states that if R is a semiprimary ring and M is an R module, the three module conditions Noetherian
Noetherian module
In abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....
, Artinian
Artinian module
In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...
and "has a composition series
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 naturally occurring modules are not semisimple, hence...
" 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
Charles Hopkins
Charles Hopkins may refer to:*Charles F. Hopkins, Union Civil War solder and winner of the Medal of Honor*Charles Jerome Hopkins*Charles Hopkins , first husband of Eliza Poe...
and a paper by Jacob Levitzki
Jacob Levitzki
Jacob Levitzki, also known as Yaakov Levitsky was an Israeli mathematician.-Biography:Levitzki was born in 1904 in Ukraine, then part of the Russian Empire, and emigrated to then Ottoman-ruled Palestine in 1912...
, 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 .
Since it is known that right Artinian rings
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
are semiprimary, a direct corollary of the theorem is: a right Artinian ring is also right Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
. 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.