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Total ring of fractions

## Summary

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. If the homomorphism from R to the new ring is to be injective, no further elements can be given an inverse.

## 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

• For a product ring A × B, the total quotient ring Q(A × B) 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.
• 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, ${\displaystyle Q(R)=R}$ .
• In a commutative von Neumann regular ring R, the same thing happens. 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, ${\displaystyle Q(R)=R}$ .

## The total ring of fractions of a reduced ring

Proposition — Let A be a reduced ring that has only finitely minimal prime ideals, ${\displaystyle {\mathfrak {p}}_{1},\dots ,{\mathfrak {p}}_{r}}$ . Then

${\displaystyle Q(A)\simeq \prod _{i=1}^{r}Q(A/{\mathfrak {p}}_{i}).}$

Geometrically, ${\displaystyle \operatorname {Spec} (Q(A))}$  is the Artinian scheme consisting (as a finite set) of the generic points of the irreducible components of ${\displaystyle \operatorname {Spec} (A)}$ .

Proof: Every element of Q(A) is either a unit or a zerodivisor. Thus, any proper ideal I of Q(A) is contained in the set of zerodivisors of Q(A); that set equals the union of the minimal prime ideals ${\displaystyle {\mathfrak {p}}_{i}Q(A)}$  since Q(A) is reduced. By prime avoidance, I must be contained in some ${\displaystyle {\mathfrak {p}}_{i}Q(A)}$ . Hence, the ideals ${\displaystyle {\mathfrak {p}}_{i}Q(A)}$  are maximal ideals of Q(A). Also, their intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A),

${\displaystyle Q(A)\simeq \prod _{i}Q(A)/{\mathfrak {p}}_{i}Q(A)}$ .

Let S be the multiplicatively closed set of non-zerodivisors of A. By exactness of localization,

${\displaystyle Q(A)/{\mathfrak {p}}_{i}Q(A)=A[S^{-1}]/{\mathfrak {p}}_{i}A[S^{-1}]=(A/{\mathfrak {p}}_{i})[S^{-1}]}$ ,

which is already a field and so must be ${\displaystyle Q(A/{\mathfrak {p}}_{i})}$ . ${\displaystyle \square }$

## Generalization

If ${\displaystyle R}$  is a commutative ring and ${\displaystyle S}$  is any multiplicatively closed set 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.

## Citations

1. ^ Matsumura 1980, p. 12.
2. ^ Matsumura 1989, p. 21.

## References

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