Total quotient ring
Encyclopedia
In abstract algebra
, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions
of a domain
to commutative ring
s that may have zero divisor
s. 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. In light of this, the total ring of quotients is optimal in the sense that "everything that could have an inverse gets an inverse".
be a commutative ring and let 
be the set of elements which are not zero divisors in 
; then 
is a multiplicatively closed set
that does not contain zero. 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 
.
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 function
s on an open set D of complex numbers is the ring of meromorphic function
s 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,
.
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.
is a commutative ring and 
any multiplicative submagma
of
with unit, one can construct the 
in a similar fashion, where only elements of 
are possible denominators. If 
, then 
is the trivial ring. For details, see Localization of a ring
.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
of a domain
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
to commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s that may have zero divisor
Zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s. 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. 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 be a commutative ring and let be the set of elements which are not zero divisors in ; then is a multiplicatively closed setMultiplicatively closed set
In abstract algebra, a subset of a ring is said to be multiplicatively closed if it is closed under multiplication and contains 1 but doesn't contain 0...
that does not contain zero. Hence we may localize
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
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 function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s on an open set D of complex numbers is the ring of meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s on D, even if D is not connected.
In an Artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
, 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
Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...
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,
.Applications
In algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
one considers a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of total quotient rings on a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, and this may be used to give one possible definition of a Cartier divisor.
Generalization
If is a commutative ring and any multiplicative submagmaMagma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
of
with unit, one can construct the in a similar fashion, where only elements of are possible denominators. If , then is the trivial ring. For details, see Localization of a ringLocalization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
.