# Total ring of fractions

In abstract algebra, the **total quotient ring**,[1] or **total ring of fractions**,[2] 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 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 .

## Examples

The total quotient ring of a product ring is the product of total quotient rings . In particular, if *A* and *B* are integral domains, it is the product of quotient fields.

The total quotient ring of the ring of holomorphic functions on an open set *D* of complex numbers 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, , and so . But since all these elements already have inverses, .

The same thing happens in a commutative von Neumann regular ring *R*. 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, .

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

## The total ring of fractions of a reduced ring

There is an important fact:

**Proposition** — Let *A* be a Noetherian reduced ring with the minimal prime ideals . Then

Geometrically, is the Artinian scheme consisting (as a finite set) of the generic points of the irreducible components of .

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 as *Q*(*A*) is reduced, by prime avoidance, *I* must be contained in some . Hence, the ideals are the maximal ideals of *Q*(*A*), whose intersection is zero. Thus, by the Chinese remainder theorem applied to *Q*(*A*), we have:

- .

Finally, is the residue field of . 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 .

## Generalization

If is a commutative ring and is any multiplicative subset in , the localization can still be constructed, but the ring homomorphism from to might fail to be injective. For example, if , then is the trivial ring.

## Notes

- Matsumura (1980), p. 12
- Matsumura (1989), p. 21

## References

- Hideyuki Matsumura,
*Commutative algebra*, 1980 - Hideyuki Matsumura,
*Commutative ring theory*, 1989