Generating set of a module: Difference between revisions

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{{Histmerge|Minimal generating set|reason=The contents of the source page were moved to here when this article was created. All but the last two edits to the source page should be merged here.}}

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