|
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated ''R''-module also may be called a finite ''R''-module or finite over ''R''.〔For example, Matsumura uses this terminology.〕 Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. ==Formal definition== The left ''R''-module ''M'' is finitely generated if there exist ''a''1, ''a''2, ..., ''a''''n'' in ''M'' such that for all ''x'' in ''M'', there exist ''r''1, ''r''2, ..., ''r''''n'' in ''R'' with ''x'' = ''r''1''a''1 + ''r''2''a''2 + ... + ''r''''n''''a''''n''. The set is referred to as a generating set for ''M'' in this case. The finite generators need not be a basis, since they need not be linearly independent over ''R''. What is true is: ''M'' is finitely generated if and only if there is a surjective ''R''-linear map: : for some ''n'' (''M'' is a quotient of a free module of finite rank.) If a set ''S'' generates a module that is finitely generated, then the finite generators of the module can be taken from ''S'' at the expense of possibly increasing the number of the generators (since only finitely many elements in ''S'' are needed to express the finite generators). In the case where the module ''M'' is a vector space over a field ''R'', and the generating set is linearly independent, ''n'' is ''well-defined'' and is referred to as the dimension of ''M'' (''well-defined'' means that any linearly independent generating set has ''n'' elements: this is the dimension theorem for vector spaces). Any module is a union of an increasing chain of finitely generated submodules. A module ''M'' is finitely generated if and only if any increasing chain ''M''''i'' of submodules with union ''M'' stabilizes: i.e., there is some ''i'' such that ''M''''i'' = ''M''. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module ''M'' is called a Noetherian 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「Finitely generated module」の詳細全文を読む スポンサード リンク
|