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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 Primary ideal ： ウィキペディア英語版 Primary ideal In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y''''n'' is also an element of ''Q'', for some ''n>0''. For example, in the ring of integers Z, (''p''n) is a primary ideal if ''p'' is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently,〔To be precise, one usually uses this fact to prove the theorem.〕 an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist〔See the references to Chatters-Hajarnavis, Goldman, Gorton-Heatherly, and Lesieur-Croisot.〕 but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity. ==Examples and properties== * The definition can be rephrased in a more symmetric manner: an ideal $\backslash mathfrak$ is primary if, whenever $x\; y\; \backslash in\; \backslash mathfrak$, we have either $x\; \backslash in\; \backslash mathfrak$ or $y\; \backslash in\; \backslash mathfrak$ or $x,\; y\; \backslash in\; \backslash sqrt\}$ denotes the radical of $\backslash mathfrak$.) * An ideal ''Q'' of ''R'' is primary if and only if every zerodivisor in ''R/Q'' is nilpotent. (Compare this to the case of prime ideals, where ''P'' is prime if every zerodivisor in ''R/P'' is actually zero.) * Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime. * Every primary ideal is primal.〔For the proof of the second part see the article of Fuchs〕 * If ''Q'' is a primary ideal, then the radical of ''Q'' is necessarily a prime ideal ''P'', and this ideal is called the associated prime ideal of ''Q''. In this situation, ''Q'' is said to be ''P''-primary. * * On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if $R\; =\; k()/(x\; y\; -\; z^2)$, $\backslash mathfrak\; =\; (\backslash overline,\; \backslash overline)$, and $\backslash mathfrak\; =\; \backslash mathfrak^2$, then $\backslash mathfrak$ is prime and $\backslash sqrt$, but we have $\backslash overline\; \backslash overline\; =\; ^2\; =\; \backslash mathfrak$, $\backslash overline\; \backslash not\; \backslash in\; \backslash mathfrak$, and $$ for all n > 0, so $\backslash mathfrak$ is not primary. The primary decomposition of $\backslash mathfrak$ is $(\backslash overline)\; \backslash cap\; (\; \backslash overline,\; \backslash overline)$; here $(\backslash overline)$ is $\backslash mathfrak$-primary and $(\; \backslash overline,\; \backslash overline)$ is $(\backslash overline,\; \backslash overline,\; \backslash overline)$-primary. * * An ideal whose radical is ''maximal'', however, is primary. * If ''P'' is a maximal prime ideal, then any ideal containing a power of ''P'' is ''P''-primary. Not all ''P''-primary ideals need be powers of ''P''; for example the ideal (''x'', ''y''2) is ''P''-primary for the ideal ''P'' = (''x'', ''y'') in the ring ''k''(), but is not a power of ''P''. * In general powers of a prime ideal ''P'' need not be ''P''-primary. (An example is given by taking ''R'' to be the ring ''k''()/(''xy'' − ''z''2), with ''P'' the prime ideal (''x'', ''z''). If ''Q'' = ''P''2, then ''xy'' ∈ ''Q'', but ''x'' is not in ''Q'' and ''y'' is not in the radical ''P'' of ''Q'', so ''Q'' is not ''P''-primary.) However every ideal ''Q'' with radical ''P'' is contained in a smallest ''P''-primary ideal, consisting of all elements ''a'' such that ''ax'' is in ''Q'' for some ''x'' not in ''P''. In particular there is a smallest ''P''-primary ideal containing ''P''''n'', called the ''n''th symbolic power of ''P''. * If ''A'' is a Noetherian ring and ''P'' a prime ideal, then the kernel of $A\; \backslash to\; A\_P$, the map from ''A'' to the localization of ''A'' at ''P'', is the intersection of all ''P''-primary ideals.〔Atiyah-Macdonald, Corollary 10.21〕 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「Primary ideal」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.