Generating set of a module: Difference between revisions

Revision as of 20:44, 5 June 2016

Revision as of 20:46, 5 June 2016

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Put in another way, there is a surjection

: <math> \bigoplus_{g \in G} R \to M, \, (r|g \in G)r_g \to rg.</math>

where we wrote (''r''|_{''g'' ∈ ''G'')} for an element in the ''g''-th component of ''G''the direct sum. (Coincidentally, since a generating set always exists; for example, ''M'' itself, this shows that a module is a quotient of a free module, a 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>