3 edited tags
2 corrected Hopkins-Levitski and some typos

It is well known that for rings, Noetherain Artinian implies Artianian Noetherian (the famous HopkinsāLevitzki theorem) and it is also well known that there are Artinian modules which are not NoeterianNoetherian. A simple example can be found in

http://en.wikipedia.org/wiki/Artinian_module#Relation_to_the_Noetherian_condition

Since rings are always finitely generated modules over themselves (all rings considered are unital), it seemed natural to me to ask whether there are finitely generated modules, which are Artinian but not Noetherian (the example given in the reference is clearly not finitely generated). I guess that if the statement "every finitely generated artinian module is noetherian" was true, I would have seen it in any standard text book on algebra, and since I haven't, I guess it's not. But still, I can't find a counter-example for this. Perhaps I'm missing something completely trivial here. I will be happy to see an example of such module or a proof that there are no (just a reference will be much appreciated too of course).

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# Are there any finitely generated artinian modules that are not notherian?

It is well known that for rings, Noetherain implies Artianian (the famous HopkinsāLevitzki theorem) and it is also well known that there are Artinian modules which are not Noeterian. A simple example can be found in

http://en.wikipedia.org/wiki/Artinian_module#Relation_to_the_Noetherian_condition

Since rings are always finitely generated modules over themselves (all rings considered are unital), it seemed natural to me to ask whether there are finitely generated modules, which are Artinian but not Noetherian (the example given in the reference is clearly not finitely generated). I guess that if the statement "every finitely generated artinian module is noetherian" was true, I would have seen it in any standard text book on algebra, and since I haven't, I guess it's not. But still, I can't find a counter-example for this. Perhaps I'm missing something completely trivial here. I will be happy to see an example of such module or a proof that there are no (just a reference will be much appreciated too of course).