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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Artin–Tate lemma ： ウィキペディア英語版 Artin–Tate lemma In algebra, the Artin–Tate lemma states: :Let ''A'' be a Noetherian ring and $B\; \backslash sub\; C$ algebras over ''A''. If ''C'' is of finite type over ''A'' and if ''C'' is finite over ''B'', then ''B'' is of finite type over ''A''. (Here, "of finite type" means "finitely generated algebra" and "finite" means "finitely generated module".) The lemma was introduced by E. Artin and J. Tate in 1951〔E Artin, J.T Tate, "A note on finite ring extensions," J. Math. Soc Japan, Volume 3, 1951, pp. 74–77〕 to give a proof of Hilbert's Nullstellensatz. == Proof == The following proof can be found in Atiyah–MacDonald. Let $x\_1,\; ...,\; x\_m$ generate $C$ as an $A$-algebra and let $y\_1,\; ...,\; y\_n$ generate $C$ as a $B$-module. Then we can write $x\_i\; =\; \backslash sum\_j\; b\_y\_j$ and $y\_iy\_j\; =\; \backslash sum\_b\_y\_k$ with $b\_,b\_\; \backslash in\; B$. Then $C$ is finite over the $A$-algebra $B\_0$ generated by the $b\_,b\_$. Using that $A$ and hence $B\_0$ is Noetherian, also $B$ is finite over $B\_0$. Since $B\_0$ is a finitely generated $A$-algebra, also $B$ is a finitely genrated $A$-algebra. 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「Artin–Tate lemma」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.