Generating set of a module

In mathematics, a generating set "G" of a module "M" over a ring "R" is a subset of "M" such that the smallest submodule of "M" containing "G" is "M" itself (the smallest submodule containing a subset is the intersection of all submodules containing the set). The set "G" is then said to generate "M". For example, the ring "R" is generated by the identity element 1 as a left "R"-module over itself. If there is a finite generating set, then a module is said to be finitely generated.
This applies to ideals, which are the submodules of the ring itself. In particular, a principal ideal is an ideal that has a generating set consisting of a single element.
Explicitly, if "G" is a generating set of a module "M", then every element of "M" is a (finite) "R"-linear combination of some elements of "G"; i.e., for each "x" in "M", there are "r"1, ..., "r""m" in "R" and "g"1, ..., "g""m" in "G" such that
Put in another way, there is a surjection
where we wrote "r""g" for an element in the "g"-th component of the direct sum. (Coincidentally, since a generating set always exists; for example, "M" itself, this shows that a module is a quotient of a free module, a useful fact.)
A generating set of a module is said to be minimal if no proper subset of the set generates the module. If "R" is a field, then a minimal generating set is the same thing as a basis. Unless the module is finitely-generated, there may exist no minimal generating set.
The cardinality of a minimal generating set need not be an invariant of the module; Z is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set }. What is uniquely determined by a module is the infimum of the numbers of the generators of the module.
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 formula_3. If "M" is flat, then this minimal generating set is linearly independent (so "M" is free). See also: minimal resolution.
A more refined information is obtained if one considers the relations between the generators; cf. free presentation of a module.