Total ring of fractions
In abstract algebra, the total quotient ring, or total ring of fractions, 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. If the homomorphism from "R" to the new ring is to be injective, no further elements can be given an inverse. Definition. Let formula_1 be a commutative ring and let formula_2 be the set of elements which are  not zero divisors in formula_1; then formula_2 is a multiplicatively closed set. Hence we may localize the ring formula_1 at the set formula_2 to obtain the total quotient ring formula_7. If formula_1 is a domain, then formula_9 and the total quotient ring is the same as the field of fractions. This justifies the notation formula_10, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain. Since formula_2 in the construction contains no zero divisors, the natural map formula_12 is injective, so the total quotient ring is an extension  of formula_1. The total ring of fractions of a reduced ring. There is an important fact: Proof: Every element of "Q"("A") is either a unit or a zerodivisor. Thus, any proper ideal "I" of "Q"("A") must consist of zerodivisors. Since the set of zerodivisors of "Q"("A") is the union of the minimal prime ideals formula_18 as "Q"("A") is reduced, by prime avoidance, "I" must be contained in some formula_18. Hence, the ideals formula_18 are the maximal ideals of "Q"("A"), whose intersection is zero. Thus, by the Chinese remainder theorem applied to "Q"("A"), we have: Finally, formula_22 is the residue field  of formula_23. Indeed, writing "S" for the multiplicatively closed set of non-zerodivisors, by the exactness of localization, which is already a field and so must be formula_25. formula_26 Generalization. If formula_1 is a commutative ring and formula_2 is any multiplicatively closed set in formula_1, the localization formula_30 can still be constructed, but the ring homomorphism from formula_1 to formula_30 might fail to be injective. For example, if formula_33, then formula_30 is the trivial ring.