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Let R be a ring with identity. An R-module M is Artinian if it satisfies the descending chain condition on submodules. What is an example of an Artinian module with a proper submodule that is not finitely generated?

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Take a look at http://en.wikipedia.org/wiki/Artinian_module#Relation_to_the_Noetherian_condition.

There you can find a concrete example of an Artinian module over the integers which is non-Noetherian and hence has a non-finitely generated submodule.

In the example at hand, the module itself is not finitely generated, and there are also lots of non-finitely generated proper submodules.

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    I just wanted to point out here a fantastic theorem due to D.D. Anderson that every Artinian module, over any commutative ring with unity, is countably generated ... http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-583cfb09-a56c-4bc4-a505-6dd90d25e4b9/c/cm38_1_02.pdf2018-08-19