Generating set of a module: Difference between revisions

Revision as of 22:50, 4 June 2016

Revision as of 22:51, 4 June 2016

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

A generating set of a module is said to be '''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>