Generating set of a module

In algebra, 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 G exists; it is the intersection of all submodules containing G). The set G is then said to generate M. For example, when the ring is viewed as a left module over itself, then R is generated by the identity element 1 as a left R-module. If there is a finite generating set, then a module is said to be finitely generated.

Explicitly, if G is a generating set of a module M, then every element of M is a (finite) R-linear combination of the elements of G; i.e., for each x in M, there are r₁, ..., r_m in R and g₁, ..., g_m such that

$x=r_{1}g_{1}+\dots +r_{m}g_{m}$.

Put in another way, there is a surjection $\oplus _{g\in G}R\to M$.

A generating set of a module is minimal if no proper subset of the set generares the module. If R is a field, then it is the same thing as a basis. Unless the module is finitely-generated, there may exist no minimal generating set.

The cardinarity 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 cardinarity is $\dim _{k}M/mM=\dim _{k}M\otimes _{R}k$. If M is flat, then this minimal generating set is linearly independent (so M is free). See also: minimal resolution.

See also

• Invariant basis number
• Flat module

References

• Dummit, David; Foote, Richard. Abstract Algebra.