Certain Artinian Rings are Noetherian

Abstract

Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

Structure of Some Noetherian Injective Modules

“…Proof. Let l(Qo) = P. Observe that in the exact sequence 0 -> P'Q/P'+iQ -> Q/P i+1 Q -> 0/P*Q -» 0, 12,6]. A simple induction and the fact that P i + 1 Q = 0 completes the proof.…”

Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki theorem (I)

“…If, in addition G and L are both $T-Artinian and L satisfies {$)% then L is #"-Noetherian. (4) We need hardly comment that Theorem 4-9 can be applied to give the results o f [ l ] , [7], [8], [10]. (5) In our forthcoming paper we shall apply 4-9 when 3C is the Serre class ^f a of lattices with Krull dimension < a, for an ordinal a ^ 0.…”

“…The Hopkins-Levitzki Theorem, discovered independently in 1939 by C. Hopkins and J. Levitzki states that a right Artinian ring with identity is right Noetherian. In the 1970s and 1980s it has been generalized to modules over non-unital rings by Shock [10], to modules satisfying the descending chain condition relative to a heriditary torsion theory by Miller-Teply [7], to Grothendieck categories by Nastasescu [8], and to upper continuous modular lattices by Albu [1]. The importance of the relative Hopkins-Levitzki Theorem in investigating the structure of some relevant classes of modules, including injectives as well as projectives is revealed in [3] and [6], where the main body of both these monographs deals with this topic.…”