Recently, I have been thinking about a problem in which I try to use the information I have about the set of generators of a finitely generated module over a nice ring and say something about the length of that. I don't want to post the problem and spoil the joy of thinking about that. Instead, I want to ask a bit more general question.
As long as I know, in general, it is not possible to look at the minimal set of generators and conclude something about the length. It is quite easy to make a multitude of counterexamples.
So, I would like to ask this question:
What are the weakest conditions we should put on the ring and the module in order to be able to use the information about the minimal set of generators and say something interesting about the length of the module?
Thanks for your help in advance.
If $A$ has finite length and $M$ has $n$ generators, then $\ell(M) \leq n \ell(A)$.
It's hard to say much more, as you probably know. E.g. already over $\mathbb Z$ there are cyclic modules (so one generator) of arbitrary length.