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''). The set ''G'' is then said to generate ''M''. 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]].

Explicitly, if ''G'' is a generating set of a module ''M'', then every element of ''M'' is a (finite) ''R''-linear combination of the elements of ''G''; i.e., for each ''x'' in ''M'', there are ''r''₁, ..., ''r''_m in ''R'' and ''g''₁, ..., ''g''_m such that

:<math>x = r_1 x_1 + \dots + r_m x_m</math>.

Put in another way, there is a surjection <math>\oplus_{g \in G} R \to M</math>.

A generating set of a module is minimal if no proper subset of the set generares 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>