Rank of An Abelian Group - Generalization

Generalization

The notion of rank can be generalized for any module M over an integral domain R, as the dimension over R₀, the quotient field, of the tensor product of the module with the field:

It makes sense, since R₀ is a field, and thus any module (or, to be more specific, vector space) over it is free.

It is a generalization, since any abelian group is a module over the integers. It easily follows that the dimension of the product over Q is the cardinality of maximal linearly independent subset, since for any torsion element x and any rational q