# Total quotient ring

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. 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 ${\displaystyle R}$ be a commutative ring and let ${\displaystyle S}$ be the set of elements which are not zero divisors in ${\displaystyle R}$; then ${\displaystyle S}$ is a multiplicatively closed set. Hence we may localize the ring ${\displaystyle R}$ at the set ${\displaystyle S}$ to obtain the total quotient ring ${\displaystyle S^{-1}R=Q(R)}$.

If ${\displaystyle R}$ is a domain, then ${\displaystyle S=R-\{0\}}$ and the total quotient ring is the same as the field of fractions. This justifies the notation ${\displaystyle Q(R)}$, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.

Since ${\displaystyle S}$ in the construction contains no zero divisors, the natural map ${\displaystyle R\to Q(R)}$ is injective, so the total quotient ring is an extension of ${\displaystyle R}$.

## Examples

The total quotient ring ${\displaystyle Q(A\times B)}$ of a product ring is the product of total quotient rings ${\displaystyle Q(A)\times Q(B)}$. 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, ${\displaystyle R^{\times }}$, and so ${\displaystyle Q(R)=(R^{\times })^{-1}R}$. But since all these elements already have inverses, ${\displaystyle Q(R)=R}$.

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, ${\displaystyle Q(R)=R}$.

## Applications

• The rational functions over a ring R{{ safesubst:#invoke:Unsubst||$N=Dubious |date=__DATE__ |$B=

{{#invoke:Category handler|main}}[dubious ] }} can be constructed from the polynomial ring R[x] as a total quotient ring.[3]

## Generalization

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

## Notes

1. Matsumura (1980), p. 12
2. Matsumura (1989), p. 21
3. {{#invoke:citation/CS1|citation |CitationClass=citation }}.

## References

• Hideyuki Matsumura, Commutative algebra, 1980
• Hideyuki Matsumura, Commutative ring theory, 1989