Dual generalizations of the Artinian and Noetherian conditions

Robert C. Shock

Vol. 54 (1974), No. 2, 227–235

DOI: 10.2140/pjm.1974.54.227

Abstract

An Artinian module can be characterized in terms of certain properties of its factor modules. A module M is Artinian if and only if the following two conditions hold for M:

(I) Every nonzero factor module of M contains a minimal submodule.

(A) The socle of every factor module of M is finitely generated

The dual to the factor module is the submodule. We state the dual of (I):

(II) Every nonzero submodule of M contains a maximal sub-module.

We call a module with property (II) a Max module and one with property (I) a Min module. Every Noetherian module is a Max module but not conversely. This paper investigates these generalizations of the Artinian and Noetherian conditions and the relationships among them.

Mathematical Subject Classification

Primary: 16A46

Milestones

Received: 26 October 1973

Published: 1 October 1974