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In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by Krull's principal ideal theorem ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''. The appellation ''regular'' is justified by the geometric meaning. A point ''x'' on an algebraic variety ''X'' is nonsingular if and only if the local ring of germs at ''x'' is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings.〔A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.〕 For Noetherian local rings, there is the following chain of inclusions: ==Characterizations== There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if is a Noetherian local ring with maximal ideal , then the following are equivalent definitions * Let where is chosen as small as possible. Then is regular if ::, :where the dimension is the Krull dimension. The minimal set of generators of are then called a ''regular system of parameters''. * Let be the residue field of . Then is regular if ::, :where the second dimension is the Krull dimension. * Let be the global dimension of (i.e., the supremum of the projective dimensions of all -modules.) Then is regular if ::, :in which case, . 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「regular local ring」の詳細全文を読む スポンサード リンク
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