Total quotient ring

In mathematics, the total quotient ring is a construction that generalizes the notion of the field of fractions of a domain to rings that may have zero divisors. The idea is to formally invert as many elements of the ring as possible without trivializing the ring.

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 that does not contain zero. 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 $R\; o\; Q(R)$ is injective, so the total quotient ring is an extension of $R$.

Examples

The total quotient ring of the ring of holomorphic functions is the ring of meromorphic functions.

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

Applications

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 $S$ any multiplicative submagma of $R$ with unit, one can construct the $S^\{-1\}R$ in a similar fashion, where only elements of $S$ are possible denominators. If $0\; in\; S$, then $S^\{-1\}R$ is the trivial ring. For details, see Localization of a ring.