Minimal generating set

In algebra, a minimal generating set of a module over a ring R is a generating set of the module such that no proper subset of the set generates 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.^[1]

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 }. (Consequently one usually considers 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

1. "ac.commutative algebra - Existence of a minimal generating set of a module - MathOverflow". mathoverflow.net.

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