# Generating set of a module

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{{Short description|Concept in mathematics}}In [mathematics](/source/mathematics), a '''generating set''' Γ of a [module](/source/module_(mathematics)) ''M'' over a [ring](/source/ring_(mathematics)) ''R'' is a [subset](/source/subset) of ''M'' such that the smallest [submodule](/source/submodule) of ''M'' containing Γ is ''M'' itself (the smallest submodule containing a subset is the [intersection](/source/intersection_(set_theory)) 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](/source/finite_set) generating set, then a module is said to be [finitely generated](/source/finitely_generated_module).

This applies to [ideals](/source/ideal_(ring_theory)), which are the submodules of the ring itself. In particular, a [principal ideal](/source/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 ''r''<sub>1</sub>, ..., ''r''<sub>''m''</sub> in ''R'' and ''g''<sub>1</sub>, ..., ''g''<sub>''m''</sub> in Γ such that

: <math>x = r_1 g_1 + \cdots + r_m g_m.</math>

Put in another way, there is a [surjection](/source/surjection)

: <math>\bigoplus_{g \in \Gamma} R \to M, \, r_g \mapsto r_g g,</math>

where we wrote ''r''<sub>''g''</sub> 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](/source/quotient_module) of a [free module](/source/free_module), a useful fact.)

A generating set of a module is said to be '''minimal''' if no [proper subset](/source/proper_subset) of the set generates the module. If ''R'' is a [field](/source/field_(mathematics)), then a minimal generating set is the same thing as a [basis](/source/basis_(linear_algebra)). Unless the module is [finitely generated](/source/finitely_generated_module), there may exist no minimal generating set.<ref>{{cite web|url=https://mathoverflow.net/q/33540 |title=ac.commutative algebra – Existence of a minimal generating set of a module – MathOverflow|work=mathoverflow.net}}</ref>

The [cardinality](/source/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 {{nowrap|{2, 3}}}. What ''is'' uniquely determined by a module is the [infimum](/source/infimum) of the numbers of the generators of the module.

Let ''R'' be a [local ring](/source/local_ring) with [maximal ideal](/source/maximal_ideal) ''m'' and [residue field](/source/residue_field) ''k'' and ''M'' finitely generated module. Then [Nakayama's lemma](/source/Nakayama's_lemma) says that ''M'' has a minimal generating set whose cardinality is <math>\dim_k M / mM = \dim_k M \otimes_R k</math>. If ''M'' is [flat](/source/flat_module), then this minimal generating set is [linearly independent](/source/linearly_independent) (so ''M'' is free). See also: [Minimal resolution](/source/minimal_resolution_(algebra)).

A more refined information is obtained if one considers the relations between the generators; see [Free presentation of a module](/source/Free_presentation_of_a_module).

== See also ==
*[Countably generated module](/source/Countably_generated_module)
*[Flat module](/source/Flat_module)<!-- explain how to use "flat" to show a minimal generating set is linearly indep. -->
*[Invariant basis number](/source/Invariant_basis_number)

== References ==
{{reflist}}
*Dummit, David; Foote, Richard. ''Abstract Algebra''.

Category:Abstract algebra

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