Explicitly, if Γ is a generating set of a module M, then every element of M is a (finite) R-linear combination of some elements of Γ; i.e., for each x in M, there are r1, ..., rm in R and g1, ..., gm in Γ such that Put in another way, there is a surjection where we wrote rg for an element in the g-th component of the direct sum.
If R is a field, then a minimal generating set is the same thing as a basis.
Let R be a local ring with maximal ideal m and residue field k and M finitely generated module.
Then Nakayama's lemma says that M has a minimal generating set whose cardinality is
If M is flat, then this minimal generating set is linearly independent (so M is free).
A more refined information is obtained if one considers the relations between the generators; see Free presentation of a module.