翻訳と辞書 Words near each other ・ G-class destroyer ・ G-class destroyer (1944) ・ G-class frigate ・ G-class landing craft ・ G-class submarine ・ G-Cloud ・ G-cluster Global Corporation ・ G-code ・ G-code (disambiguation) ・ G-Collections ・ G-CSF factor stem-loop destabilising element ・ G-Darius ・ G-Day ・ G-Day (disambiguation) ・ G-DOC ・ G-domain ・ G-Dragon ・ G-Dragon discography ・ G-Eazy ・ G-Enka ・ G-expectation ・ G-factor (physics) ・ G-Fest ・ G-flat ・ G-flat major ・ G-force ・ G-Force (album) ・ G-Force (film) ・ G-Force (video game) ・ G-Force GF09 Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク G-domain ： ウィキペディア英語版 G-domain In mathematics, an integral domain $D$ is a G-domain if and only if: # Its quotient field is a simple extension of $D$ # Its quotient field is a finite extension of $D$ # Intersection of its nonzero prime ideals (not to be confused with nilradical) is nonzero # There is an element $u$ such that for any nonzero ideal $I$, $u^n\backslash in\; I$ for some $n$.〔Kaplansky, Irving. ''Commutative Algebra''. Polygonal Publishing House, 1974, p. 12,13.〕 A G-ideal is defined as an ideal $I\backslash subset\; R$ such that $R/I$ is a G-domain. Since a factor ring is an integral domain if and only if the ring is factored by a prime ideal, every G-ideal is also a prime ideal. G-ideals can be used as a refined collection of prime ideals in the following sense: Radical can be characterized as the intersection of all prime ideals containing the ideal, and in fact we still get the radical even if we take the intersection over the G-ideals.〔Kaplansky, Irving. ''Commutative Algebra''. Polygonal Publishing House, 1974, p. 16,17.〕 Every maximal ideal is a G-ideal, since quotient by maximal ideal is a field, and a field is trivially a G-domain. Therefore, maximal ideals are G-ideals, and G-ideals are prime ideals. G-ideals are the only maximal ideals in Jacobson ring, and in fact this is an equivalent characterization of a Jacobson ring: a ring is a Jacobson ring when all maximal ideals are G-ideals. This leads to a simplified proof of the Nullstellensatz.〔Kaplansky, Irving. ''Commutative Algebra''. Polygonal Publishing House, 1974, p. 19.〕 It is known that given $T\backslash supset\; R$, a ring extension of a G-domain, $T$ is algebraic over $R$ if and only if every ring extension between $T$ and $R$ is a G-domain.〔Dobbs, David. "G-Domain Pairs". Trends in Commutative Algebra Research, Nova Science Publishers, 2003, p. 71-75.〕 A Noetherian domain is a G-domain iff its rank is at most one, and has only finitely many maximal ideals (or equivalently, prime ideals).〔Kaplansky, Irving. "Commutative Algebra''. Polygonal Publishing House, 1974, p. 19.〕 ==References== 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「G-domain」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.