Generating set of a module: Difference between revisions

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{{Short description|Concept in mathematics}}In [[abstract algebra|algebramathematics]], 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 a subset is the [[intersection (set theory)|intersection]] of all submodules containing the set). The set ''G''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 ''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 ''r''₁, ..., ''r''_''m'' in ''R'' and ''g''₁, ..., ''g''_''m'' in ''G'' such that

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

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

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

Put in another way, there is a surjection

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

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

: <math> \bigoplus_{g \in G\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; for example, ''M'' itself, this shows that a module is a quotient of a free module, a useful fact.)

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, 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 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=https://mathoverflow.net/q/33540 |title=ac.commutative algebra – Existence of a minimal generating set of a module – MathOverflow|work=mathoverflow.net}}</ref>

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 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=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.

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, then this minimal generating set is [[linearly independent]] (so ''M'' is free). See also: [[minimal resolution (algebra)|minimal resolution]].

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)|minimalMinimal resolution]].

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

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

== See also ==

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[[Category:Abstract algebra]]

{{algebra-stub}}