Generating set of a module: Difference between revisions

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{{Short description|Concept in mathematics}}In algebra[[mathematics]], a '''generating set''' ''G''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 ''G''Γ is ''M'' itself (the smallest submodule containing ''G''a exists; itsubset is the [[intersection (set theory)|intersection]] of all submodules containing ''G''the set). The set ''G''Γ 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 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 ''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''.

Explicitly, if ''G''Γ is a generating set of a module ''M'', then every element of ''M'' is a (finite) ''R''-linear combination of thesome elements of Γ; i.e., for each ''Gx'' in ''M'', there are ''r''₁, ..., ''r''_''m'' in ''R'' and ''g''₁, ..., ''g''_''m'' in Γ such that

A generating set of a module is minimal if no proper subset of the set generares the module. If ''R'' is a [[field (mathematics)|field]], then it 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=http://mathoverflow.net/questions/33540/existence-of-a-minimal-generating-set-of-a-module|title=ac.commutative algebra - Existence of a minimal generating set of a module - MathOverflow|work=mathoverflow.net}}</ref>

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

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.

Put in another way, there is a [[surjection]]

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 <math>\dim_k M / mM = \dim_k M \otimes_R k</math>. If ''M'' is flat, then this minimal generating set is [[linearly independent]] (so ''M'' is free). See also: [[minimal resolution (algebra)|minimal resolution]].

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

where we wrote ''r''_''g'' 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 generaresgenerates the module. If ''R'' is a [[field (mathematics)|field]], then ita 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=httphttps://mathoverflow.net/questionsq/33540/existence-of-a-minimal-generating-set-of-a-module |title=ac.commutative algebra -– Existence of a minimal generating set of a module -– MathOverflow|work=mathoverflow.net}}</ref>

The cardinarity[[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 cardinaritycardinality 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)|minimalMinimal resolution]].

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

== See also ==

*[[InvariantCountably basisgenerated numbermodule]]

*[[Flat module]]<!-- explain how to use "flat" to show a minimal generating set is linearly indep. -->

*[[Invariant basis number]]

== References ==

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[[Category:Abstract algebra]]

{{algebra-stub}}