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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 Associated graded ring ： ウィキペディア英語版 Associated graded ring In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring: :$\backslash operatorname\_I\; R\; =\; \backslash oplus\_^\backslash infty\; I^n/I^$. Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the graded module over $\backslash operatorname\_I\; R$: :$\backslash operatorname\_I\; M\; =\; \backslash oplus\_0^\backslash infty\; I^n\; M/\; I^\; M$. == Basic definitions and properties == For a ring ''R'' and ideal ''I'', multiplication in $\backslash operatorname\_IR$ is defined as follows: First, consider homogeneous elements $a\; \backslash in\; I^i/I^$ and $b\; \backslash in\; I^j/I^$ and suppose $a\text{'}\; \backslash in\; I^i$ is a representative of ''a'' and $b\text{'}\; \backslash in\; I^j$ is a representative of ''b''. Then define $ab$ to be the equivalence class of $a\text{'}b\text{'}$ in $I^/I^$. Note that this is well-defined modulo $I^$. Multiplication of inhomogeneous elements is defined by using the distributive property. A ring or module may be related to its associated graded through the initial form map. Let ''M'' be an ''R''-module and ''I'' an ideal of ''R''. Given $f\; \backslash in\; M$, the initial form of ''f'' in $\backslash operatorname\_I\; M$, written $\backslash mathrm(f)$, is the equivalence class of ''f'' in $I^mM/I^M$ where ''m'' is the maximum integer such that $f\backslash in\; I^mM$. If $f\; \backslash in\; I^mM$ for every ''m'', then set $\backslash mathrm(f)\; =\; 0$. The initial form map is only a map of sets and generally not a homomorphism. For a submodule $N\; \backslash subset\; M$, $\backslash mathrm(N)$ is defined to be the submodule of $\backslash operatorname\_I\; M$ generated by $\backslash$. This may not be the same as the submodule of $\backslash operatorname\_IM$ generated by the only initial forms of the generators of ''N''. A ring inherits some "good" properties from its associated graded ring. For example, if ''R'' is a noetherian local ring, and $\backslash operatorname\_I\; R$ is an integral domain, then ''R'' is itself an integral domain. 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「Associated graded ring」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.