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

Abstract

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 Năstăsescu [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. A discussion on the various forms of the Hopkins–Levitzki Theorem for modules and Grothendieck categories and the connection between them may be found in [3].