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 a domain to commutative rings that may have zero divisors. The construction embeds the ring in a larger ring, giving every non-zerodivisor of the smaller ring an inverse in the larger ring. Nothing more in the small ring can be given an inverse, because zero divisors are impossible to invert^[3]. In light of this, the total ring of quotients is optimal in the sense that "everything that could have an inverse gets an inverse".

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 $R\to Q(R)$ 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, $Q(R)=R$ .

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.

Notes

^ Matsumura (1980), p. 12
^ Matsumura (1989), p. 21
^ If one supposes a is a nonzero zero divisor in R and also a unit in its total ring of quotients Q, then ab=0 for some nonzero b in R and ca=1 for a c in Q, and then 0=c(ab)=(ca)b=b, but b was assumed to be nonzero. This contradiction shows a zero divisor of R cannot be a unit in Q.

References

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

[1] Matsumura (1980), p. 12

[2] Matsumura (1989), p. 21

[3] If one supposes a is a nonzero zero divisor in R and also a unit in its total ring of quotients Q, then ab=0 for some nonzero b in R and ca=1 for a c in Q, and then 0=c(ab)=(ca)b=b, but b was assumed to be nonzero. This contradiction shows a zero divisor of R cannot be a unit in Q.

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