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 R be a commutative ring and let S be the set of elements that 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. If R is a domain, then S = R-{0} 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 \to Q(R) is injective, so the total quotient ring is an extension of R.

Examples

The total ring of fractions of a reduced ring

Proof: Every element of Q(A) is either a unit or a zero divisor. Thus, any proper ideal I of Q(A) is contained in the set of zero divisors of Q(A); that set equals the union of the minimal prime ideals since Q(A) is reduced. By prime avoidance, I must be contained in some. Hence, the ideals are maximal ideals of Q(A). Also, their intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A), Let S be the multiplicatively closed set of non-zero-divisors of A. By exactness of localization, which is already a field and so must be. \square

Generalization

If R is a commutative ring and S is any multiplicatively closed set in R, the localization S^{-1}R can still be constructed, but the ring homomorphism from R to S^{-1}R might fail to be injective. For example, if 0 \in S, then S^{-1}R is the trivial ring.

Citations