Field of fractions

In the ratio of the ring body of the ring ( with specific properties ) a high amount of this ring, to which the addition and multiplication of the ring will continue, and having each element except for a multiplicative inverse. The most prominent example is the field of rational numbers as a quotient field of the ring of integers. A generalization of the concept of not necessarily divisors of zero rings is given by the localization.

Definition

It should be an integral domain. The smallest body in which can be embedded is called the quotient field or body of fractures of the integrity of the ring. Commonly used is the symbolic abbreviation or.

Remark

Every integral domain can be embedded in a " smallest" body. All bodies in which the integrity of the ring may be embedded, included a smallest to this body, the body of the quotient integrity range isomorphic body part.

Properties

  • Is an integral domain, then the elements of the quotient field have the form with and.
  • The quotient field of a body is up to isomorphism the body itself
  • Abstract we define the quotient field of a zero-divisor free commutative ring ( which does not necessarily have to contain a one- element ) by the following universal property: A quotient field is a pair, with a body and an injective ring homomorphism, with the property that, for each pair wherein a body and an injective ring homomorphism is an injective Körperhomomorphismus are with. This clearly means that one in which you can embed R in each body, also can embed the quotient field of R (the latter embedding a continuation of the first is ).

From the last property it follows that the smallest containing body, and that it is unique up to isomorphism (ie, it is justified to speak of " the " quotient field ).

Construction

It is the quotient field of an integral ring construct as follows:

  • Explain to an equivalence relation
  • Usually one writes for the equivalence class of.
  • It now sets the set of equivalence classes.
  • Define the addition and multiplication as follows representative example:
  • The neutral element with respect to addition is, is the neutral element with respect to multiplication.
  • For the inverse is given with respect to addition by, for the inverse with respect to multiplication is given by.
  • Thus, a body, in particular, is an injective homomorphism, which mediates the desired embedding. It is true.

For the well- definedness of the structure of the reduction rule in rings integrity is critical, ie for that from always follows. This follows easily from the zero divisor of the ring of freedom.

Examples

  • The quotient field of the ring of integers is the field of rational numbers.
  • The quotient field of the polynomial ring is the rational function field
  • The quadratic number fields is the quotient field of Gaussian integers
  • Is the integrity of the ring around the body of the functions and functions to meromorphic. By the Weierstrass product theorem one sees that one can write on any meromorphic function as a quotient of two entire functions, hence is.
Algebra Complex analysis
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