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 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.

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 r₁, ..., r_m in R and g₁, ..., g_m in G such that

x=r_{1}g_{1}+\cdots +r_{m}g_{m}.\,

Put in another way, there is a surjection

\bigoplus _{g\in G}R\to M,\,r_{g}\mapsto r_{g}g.

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

A more refined information is obtained if one considers the relations between the generators; cf. free presentation of a module.

References

↑ "ac.commutative algebra – Existence of a minimal generating set of a module – MathOverflow". mathoverflow.net.

Dummit, David; Foote, Richard. Abstract Algebra.

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