|
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a field, there exist non-free modules. Given any set , there is a free module with basis , which is called ''free module on'' or ''module of formal linear combinations'' of the elements of . ==Definition== A free module is a module with a basis: a linearly independent generating set. For an -module , the set is a basis for if: # is a generating set for ; that is to say, every element of is a finite sum of elements of multiplied by coefficients in ; # is linearly independent, that is, for distinct elements of implies that (where is the zero element of and is the zero element of ). If has invariant basis number, then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the rank of the free module . The free module is said to be ''free of rank n'', or simply ''free of finite rank'' if the cardinality is finite. Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each . 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「free module」の詳細全文を読む スポンサード リンク
|