If we look at a finite dimensional vector space over a field $F$ as a noetherian $F$-module, we can view the dimension of the vector space as the length of the maximal ascending chain of subspaces. A chain being a sequence of subspaces which contain each other, without multiplicity, e.g., $V_1 \subsetneq V_2 \subsetneq\ ...\ \subsetneq V_n$.
Can we extend the same idea to any noetherian $R$-module? Or is there an example of a noetherian $R$-module which for any $N > 0$ has a chain of length larger than $N$?