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In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have simpler structure than the general ones and Hensel's lemma applies to them. Geometrically, a completion of a commutative ring ''R'' concentrates on a formal neighborhood of a point or a Zariski closed subvariety of its spectrum Spec ''R''. == General construction == Suppose that ''E'' is an abelian group with a descending filtration : of subgroups, one defines the completion (with respect to the filtration) as the inverse limit: : This is again an abelian group. Usually ''E'' is an ''additive'' abelian group. If ''E'' has additional algebraic structure compatible with the filtration, for instance ''E'' is a filtered ring, a filtered module, or a filtered vector space, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both to commutative and noncommutative rings. As may be expected, this produces a complete topological ring. 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「Completion (algebra)」の詳細全文を読む スポンサード リンク
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