Abstract
An Artίnίan module can be characterized in terms of certain properties of its factor modules. A module Mis 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 ofM is finitely generated. The dual to the factor module is the submodule. We state the dual of (I): (Π) Every nonzero submodule ofM contains a maximal submodule. We call a module with property (Π) 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. Throughout this paper M denotes a right module over an arbitrary
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