Vol. 54, No. 2, 1974

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Dual generalizations of the Artinian and Noetherian conditions

Robert C. Shock

Vol. 54 (1974), No. 2, 227–235

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
Received: 26 October 1973
Published: 1 October 1974
Robert C. Shock