Total ring of fractions

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 → Q ( R ) is injective, so the total quotient ring is an extension of R .

Examples

The total quotient ring Q ( A × B ) of a product ring is the product of total quotient rings Q ( A ) × 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 × , and so Q ( R ) = ( R × ) − 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 .

The total ring of fractions of a reduced ring

There is an important fact:

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

Finally, Q ( A ) / p i Q ( A ) is the residue field of p i . 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 Q ( A / p i ) . ◻

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 ∈ S , then S − 1 R is the trivial ring.