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{{Short description|Concept in mathematics}}In [[mathematics]], a '''generating set''' Γ of a [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|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 (set theory)|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 set|finite]] generating set, then a module is said to be [[finitely generated module|finitely generated]].

This applies to [[ideal (ring theory)|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 ''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]]

: <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 module|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 (mathematics)|field]], then a minimal generating set is the same thing as a [[basis (linear algebra)|basis]]. Unless the module is [[finitely generated module|finitely generated]], 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]] 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]] 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>\dim_k M / mM = \dim_k M \otimes_R k</math>. If ''M'' is [[flat module|flat]], then this minimal generating set is [[linearly independent]] (so ''M'' is free). See also: [[minimal resolution (algebra)|Minimal resolution]].

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

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

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

[[Category:Abstract algebra]]


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