Answer: Look at generators!
For a vector space we can always give linearly independent
For a module, any list of generators almost always is not
independent; there are relations.
And the relations have relations and so on and so forth!
This is where homology and cohomology come from, an important
method for studying modules. For more see the following articles:
- L. Avramov, R-O Buchweitz & S. Iyengar, Class and rank of differential modules, Invent. Math. 169 (2007), 1-35.
- D. Benson, S. Iyengar & H. Krause, Stratifying modular representations of finite groups, Ann. of Math. 174 (2011) 1643--1684.
- T. Marley, joint with Janet Vassilev, Cofiniteness and associated primes of local cohomology modules, Journal of Algebra 256 (2002), 180-193.
But what about K-Theory?