it seems that a module over a noetherian ring R is finitely generated if and only if it has finite length (sorry, it turns out to be false! i must have had a misunderstanding!)
but why in the following two cases, we have some extra assumptions?
Example (i) A module M over a (commutative) noetherian ring R has finite length if and only if it is finitely generated and AssM consists of only maximal ideals, where AssM is the set of associated primes for M.
(definition: We say that a prime ideal p is an associated prime for M if there exists m in M such that p = ann m.)
Example (ii) in this thesis: http://digitalcommons.mcmaster.ca/opendissertations/3531/ Page 12 Lemma 2.15: A finitely generated $A$-module is $\Sigma$-torsion if and only if it has finite length.
As i understand: $A$ is "weyl algebra" which is noetherian. so why do we need the extra assumption "$\Sigma$-torsion"?