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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 total quotient ring ： ウィキペディア英語版 Total ring of fractions In abstract algebra, the total quotient ring,〔Matsumura (1980), p. 12〕 or total ring of fractions,〔Matsumura (1989), p. 21〕 is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ''R'' in a larger ring, giving every non-zero-divisor of ''R'' an inverse in the larger ring. Nothing more in ''A'' can be given an inverse, if one wants the homomorphism from ''A'' to the new ring to be injective. == Definition == Let $R$ be a commutative ring and let $S$ be the set of elements which are not zero divisors in $R$; then $S$ is a multiplicatively closed set. Hence we may localize the ring $R$ at the set $S$ to obtain the total quotient ring $S^R=Q(R)$. If $R$ is a domain, then $S=R-\backslash$ and the total quotient ring is the same as the field of fractions. This justifies the notation $Q(R)$, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain. Since $S$ in the construction contains no zero divisors, the natural map $R\; \backslash to\; Q(R)$ is injective, so the total quotient ring is an extension of $R$. 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「Total ring of fractions」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.