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 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 be a commutative ring and let
be the set of elements which are not zero divisors in
; then
is a multiplicatively closed set. Hence we may localize the ring
at the set
to obtain the total quotient ring
.
If is a domain, then
and the total quotient ring is the same as the field of fractions. This justifies the notation
, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.
Since in the construction contains no zero divisors, the natural map
is injective, so the total quotient ring is an extension of
.
Examples
The total quotient ring of a product ring is the product of total quotient rings
. 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, , and so
. But since all these elements already have inverses,
.
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, .
- 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.
The total ring of fractions of a reduced ring
There is an important fact:
Proposition — Let A be a Noetherian reduced ring with the minimal prime ideals . Then
Geometrically, is the Artinian scheme consisting (as a finite set) of the generic points of the irreducible components of
.
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 as Q(A) is reduced, by prime avoidance, I must be contained in some
. Hence, the ideals
are the maximal ideals of Q(A), whose intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A), we have:
.
Finally, is the residue field of
. 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 .
Generalization
If is a commutative ring and
is any multiplicative subset in
, the localization
can still be constructed, but the ring homomorphism from
to
might fail to be injective. For example, if
, then
is the trivial ring.
Notes
References
- Hideyuki Matsumura, Commutative algebra, 1980
- Hideyuki Matsumura, Commutative ring theory, 1989