Generating set of a module: Difference between revisions

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{{Short description|Concept in mathematics}}In algebra[[mathematics]], a '''minimal generating set''' Γ of a [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'' is a [[generator (mathematics)|generating setsubset]] of the module''M'' such that nothe propersmallest subset[[submodule]] of the''M'' setcontaining generatesΓ the module. Ifis ''RM'' isitself (the smallest submodule containing a subset is the [[fieldintersection (mathematicsset theory)|fieldintersection]], thenof itall issubmodules containing the sameset). thingThe asset aΓ [[basisis (linearthen algebra)|basis]]said to generate ''M''. UnlessFor example, the modulering ''R'' is [[finitely-generated module|finitely-generated]],by therethe mayidentity existelement no1 minimalas generatinga set.<ref>{{citeleft web|url=http://mathoverflow.net/questions/33540/existence-of-a-minimal-generating-set-of-a''R''-module|title=ac.commutative algebraover -itself. ExistenceIf ofthere is a minimal[[finite set|finite]] generating set, ofthen a module -is MathOverflowsaid to be [[finitely generated module|work=mathoverflowfinitely generated]].net}}</ref>

TheThis cardinarityapplies ofto a[[ideal minimal(ring generatingtheory)|ideals]], setwhich needare notthe be an invariantsubmodules of the module;ring '''Z'''itself. isIn generated asparticular, a [[principal ideal by 1, but it]] is alsoan generatedideal by,that say,has a minimal generating set { 2, 3 }. (Consequently one usually considers the [[infimum]]consisting of thea numberssingle of the generators of the moduleelement.)

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''₁, ..., ''r''_''m'' in ''R'' and ''g''₁, ..., ''g''_''m'' in Γ such that

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>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''_''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 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 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}}