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).