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, 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 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 + \dotscdots + r_m g_m. \, </math>.

Put in another way, there is a surjection

: <math> \oplus_bigoplus_{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>

The cardinality of a minimal generating set need not be an invariant of the module; '''Z''' is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set {{nowrap|{ 2, 3 }}}. What is uniquely determined by a module is the [[infimum]] of the numbers of the generators of the module.

Let ''R'' be a local ring with maximal ideal ''m'' and residue field ''k'' and ''M'' finitely generated module. Then [[Nakayama's lemma]] says that ''M'' has a minimal generating set whose cardinality is <math>\dim_k M / mM = \dim_k M \otimes_R k</math>. If ''M'' is flat, then this minimal generating set is [[linearly independent]] (so ''M'' is free). See also: [[minimal resolution (algebra)|minimal resolution]].