Generating set of a module
In mathematics, a generating set Γ of a module M over a ring R is a subset of M such that the smallest submodule of M containing Γ is M itself (the smallest submodule containing a subset is the intersection of all submodules containing the set). The set Γ 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 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
- [math]\displaystyle{ x = r_1 g_1 + \cdots + r_m g_m. }[/math]
Put in another way, there is a surjection
- [math]\displaystyle{ \bigoplus_{g \in \Gamma} R \to M, \, r_g \mapsto r_g g, }[/math]
where we wrote rg for an element in the g-th component of the direct sum. (Coincidentally, since a generating set always exists, e.g. 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.[1]
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 {2, 3}. 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 [math]\displaystyle{ \dim_k M / mM = \dim_k M \otimes_R k }[/math]. 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; see Free presentation of a module.
See also
References
- Dummit, David; Foote, Richard. Abstract Algebra.
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