Generating set of a module: Difference between revisions

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In algebra, a '''minimal generating set''' of a [[module (mathematics)|module]] over a [[ring (mathematics)|ring]] ''R'' 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.)