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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 Homogeneous element ： ウィキペディア英語版 Graded ring In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups $R\_i$ such that $R\_i\; R\_j\; \backslash subset\; R\_$. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid or group. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to a non-associative algebra as well; e.g., one can consider a graded Lie algebra. == First properties == Let :$A\; =\; \backslash bigoplus\_A\_n\; =\; A\_0\; \backslash oplus\; A\_1\; \backslash oplus\; A\_2\; \backslash oplus\; \backslash cdots$ be a graded ring. * $A\_0$ is a subring of ''A'' (in particular, the additive identity 0 and the multiplicative identity 1 are homogeneous elements of degree zero.) *A commutative $\backslash mathbb\_0$-graded ring $A=\backslash oplus\_0^\backslash infty\; A\_i$ is a Noetherian ring if and only if $A\_0$ is Noetherian and ''A'' is finitely generated as an algebra over $A\_0$. For such a ring, the generators may be taken to be homogeneous. Elements of any factor $A\_n$ of the decomposition are called homogeneous elements of degree ''n''. An ideal or other subset $\backslash mathfrak$ ⊂ ''A'' is homogeneous if, for every element ''a'' ∈ $\backslash mathfrak$, when ''a=a1+a2+...+an'' with all ''ai'' homogeneous elements, then all the ''ai'' are in the ideal. For a given ''a'' these homogeneous elements are uniquely defined and are called the homogeneous parts of ''a''. If ''I'' is a homogeneous ideal in ''A'', then $A/I$ is also a graded ring, and has decomposition :$A/I\; =\; \backslash bigoplus\_(A\_n\; +\; I)/I.$ Any (non-graded) ring ''A'' can be given a gradation by letting ''A''0 = ''A'', and ''A''''i'' = 0 for ''i'' > 0. This is called the trivial gradation on ''A''. 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「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.