Relation To The Noetherian Condition
Unlike the case of rings, there are Artinian modules which are not Noetherian modules. For example, consider the p-primary component of, that is, which is isomorphic to the p-quasicyclic group, regarded as -module. The chain does not terminate, so (and therefore ) is not Noetherian. Yet every descending chain of (without loss of generality) proper submodules terminates: Each such chain has the form for some integers ..., and the inclusion of implies that must divide . So ... is a decreasing sequence of positive integers. Thus the sequence terminates, making Artinian.
Over a commutative ring, every cyclic Artinian module is also Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable length as shown in the article of Hartley and summarized nicely in the Paul Cohn article dedicated to Hartley's memory.
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