Generating set of a module: Difference between revisions

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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 { 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 cardinaritycardinality 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]].

== See also ==