Total ring of fractions explained

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 S ^{-1} R = Q(R) .

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

• For a product ring, the total quotient ring is the product of total quotient rings . In particular, if A and B are integral domains, it is the product of quotient fields.
• For the ring of holomorphic functions on an open set D of complex numbers, the total quotient ring is the ring of meromorphic functions on D, even if D is not connected.
• In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero-divisors is the group of units of the ring,

R ^{x} , and so Q(R) = (R ^{x} ) ^{-1} R . But since all these elements already have inverses, Q(R) = R .

• In a commutative von Neumann regular ring R, the same thing happens. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a = axa for some x in R, giving the equation a(xa − 1) = 0. Since a is not a zero divisor, xa = 1, showing a is a unit. Here again,

Q(R) = R .

• In algebraic geometry one considers a sheaf of total quotient rings on a scheme, and this may be used to give the definition of a Cartier divisor.

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

ak{p} _{i} Q(A) since Q(A) is reduced. By prime avoidance, I must be contained in some ak{p} _{i} Q(A) . Hence, the ideals ak{p} _{i} Q(A) are maximal ideals of Q(A). Also, their intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A), Q(A) \simeq \prod _{i} Q(A)/ ak{p} _{i} Q(A) .Let S be the multiplicatively closed set of non-zero-divisors of A. By exactness of localization, Q(A)/ ak{p} _{i} Q(A) = A[S ^{-1} ] / ak{p} _{i} A[S ^{-1} ] = (A / ak{p} ^{-1} _{i)[S} ] ,which is already a field and so must be Q(A/ ak{p} _{i)} . \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