Blancmange curve

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. 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 $R$ be a commutative ring and let $S$ be the set of elements which 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

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

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, $Q(R)=R$.

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• 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.

Generalization

If $R$ is a commutative ring and $S$ is any multiplicative subset 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.

References

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