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"?

notto accept Dietrich's interesting comment as an answer. As your own comment to it makes evident, it does not answer your question! $\endgroup$ – Mariano Suárez-Álvarez May 30 '13 at 16:45