For an arbitrary unital, commutative ring $R$ and a corresponding unital $R$-module $M$, does there exist a topology on $M$ s.t. the induced Krull-dimension of $M$ coincides with it's length as a $R$-module?
Now more generally, can the length of a module be derived from Krull dimension somehow? Eg., is something similar possible as with $\text{dim}\left(R\right)=\text{dim}\left(\text{Spec}\left(R\right)\right)$ (using Zariski)?
Or does it eg. fail at constructing useful topologies on an arbitrary module already?
Edit: Obviously, the definition of length may be used in the construction of the topology. Is it clear what I want?