It is well known that for *rings*, Artinian implies Noetherian (the famous Hopkins–Levitzki theorem) and it is also well known that there are Artinian *modules* which are not Noetherian. 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).



Are there any finitely generated artinian modules that are not Noetherian?
Suppose you have an Artinian but not Noetherian finitely generated $R$ module $M$. Let $0\leq M\_1\leq M\_2\leq \cdots \leq M\_n=M$ be a finite chain of $R$-modules such that each composition factor $M\_i/M\_{i-1}$ is cyclic for each $i$. 


Certainly each composition is Artinian since subquotients of Artinian modules are Artinian. Also one of the composition factors must be non-Noetherian since extensions of Noetherian modules by Noetherian modules are Noetherian. Thus, we may assume that $M$ is a *cyclic* $R$-module. 


Now if $R$ is commutative, $M$ is a quotient ring $R/I$ which is Artinian as such and so Noetherian also, as you say. 


If $R$ is non-commutative then I'm not sure what the answer is. 


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Added: It seems from the wikipedia article linked from the question that Hartley showed that there are cyclic Artinian and non-Noetherian modules over certain non-commutative rings and Cohn gave another construction nearly twenty years later. See the links I give in the comments on the question for precise references.


