Generating set of a module: Difference between revisions

Revision as of 22:49, 4 June 2016

Revision as of 22:50, 4 June 2016

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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''). 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]].

Explicitly, if ''G'' is a generating set of a module ''M'', then every element of ''M'' is a (finite) ''R''-linear combination of thesome 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

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

Put in another way, there is a surjection <math>\oplus_{g \in G} R \to M</math>. (Coincidentally, since a generating set always exists, this shows that a module is a quotient of a free module, a very useful fact.)