We generalise the familiar notions of invariant basis number, rank condition, stable finiteness and strong rank condition
from rings to modules. We study the inter relationship between these properties, identify various classes of modules possessing
these properties and investigate the effect of many standard module theoretic operations on each one of these properties.
We also tackle the important problem of preservation or non-preservation of these properties when we pass respectively to
the module of polynomials, power series or inverse polynomials.