Generating set of a module: Difference between revisions

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In algebra, a '''generating set'' ''G''' 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'' is ''M'' itself (the "smallest submodule containing ''G'' exists; it is the intersection of all submodules containing ''G''). 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. If there is a finite generating set, then a module is said to be [[finitely generated module|finitely generated]].

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In algebra, a '''minimalA generating set''' ofis aminimal [[moduleif (mathematics)|module]]no overproper asubset [[ringof (mathematics)|ring]]the ''R''set is a [[generator (mathematics)|generating set]] of the module such that 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=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>

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 }. (Consequently one usually considers the [[infimum]] of the numbers of the generators of the module.)