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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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    @Jon, of course not: every module has finitely generated submodules, independently of how monstrous is the ring.2010-10-20
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    I wonder if there is an example of a finitely generated Artinian ring which is not Noetherian.2010-10-20
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    @Rasmus: Hmmm... A left Artinian ring is necessarily left Noetherian; the example in Wikipedia is of an Artinian *module* that is not Noetherian. So when you say "a concrete example of an Artinian ring which is non-Noetherian", did you mean "Artinian module"?2010-10-20
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    @Arturo Magidin: Yes, thank you!2010-10-20
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    Thanks for the quick answer, guys! @Mariano: Yep, should have typed "there exists".2010-10-20
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    @Rasmus: Nice refinement of the question!2010-10-20
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    @Rasmus: Note that this will be slightly tricky, because the ring of coefficients will have to be non-Noetherian (otherwise every f.g. module is Noetherian).2010-10-20
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    @Rasums: I think the answer is "no" if the coefficient ring is commutative.2010-10-20
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    @Rasmus; it is a theorem that every left Artinian ring is Noetherian, so there can be no finitely generated Artinian ring that is not Noetherian. Did you mean "module" again?2010-10-21
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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