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Total ring of fractions : 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 be a commutative ring and let be the set of elements which are not zero divisors in ; then is a multiplicatively closed set. Hence we may localize the ring at the set to obtain the total quotient ring . If is a domain, then and the total quotient ring is the same as the field of fractions. This justifies the notation , which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain. Since in the construction contains no zero divisors, the natural map is injective, so the total quotient ring is an extension of .
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