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In algebra, a minimal generating set of a module over a ring ''R'' is a generating set of the module such that no proper subset of the set generates the module. If ''R'' is a field, then it is the same thing as a basis. Unless the module is finitely-generated, there may exist no minimal generating set.〔(【引用サイトリンク】title=ac.commutative algebra - Existence of a minimal generating set of a module - MathOverflow )〕 The cardinarity 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 . (Consequently one usually considers 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 cardinarity is . If ''M'' is flat, then this minimal generating set is linearly independent (so ''M'' is free). See also: minimal resolution. == See also == *Invariant basis number *Flat module 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Minimal generating set」の詳細全文を読む スポンサード リンク
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