Generating set of a module: Difference between revisions

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{{Short description|InConcept in mathematics, a generating set G of a module M over a ring R is a subset of M such that the smallest submodule of M containing G is M itself. The set G is then said to generate M. For example, the ring R is generated by the identity element 1 as a}}In [[mathematics]], 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 a subset is the intersection of all submodules containing the set). The set ''G'' 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 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.