Editing Field of fractions (section)

== Definition ==

Given an integral domain <math>R</math> and letting <math>R^* = R \setminus \{0\}</math>, we define an [[equivalence relation]] on <math>R \times R^*</math> by letting <math>(n,d) \sim (m,b)</math> whenever <math>nb = md</math>. We denote the [[equivalence class]] of <math>(n,d)</math> by <math>\frac{n}{d}</math>. This notion of equivalence is motivated by the rational numbers <math>\Q</math>, which have the same property with respect to the underlying [[ring (mathematics)|ring]] <math>\Z</math> of integers.

Then the '''field of fractions''' is the set <math>\text{Frac}(R) = (R \times R^*)/\sim</math> with addition given by
:<math>\frac{n}{d} + \frac{m}{b} = \frac{nb+md}{db}</math>  
and multiplication given by 
:<math>\frac{n}{d} \cdot \frac{m}{b} = \frac{nm}{db}.</math>
One may check that these operations are well-defined and that, for any integral domain <math>R</math>, <math>\text{Frac}(R)</math> is indeed a field. In particular, for <math>n,d \neq 0</math>, the multiplicative inverse of <math>\frac{n}{d}</math> is as expected: <math>\frac{d}{n} \cdot \frac{n}{d} = 1</math>.

The embedding of <math>R</math> in <math>\operatorname{Frac}(R)</math> maps each <math>n</math> in <math>R</math> to the fraction <math>\frac{en}{e}</math> for any nonzero <math>e\in R</math> (the equivalence class is independent of the choice <math>e</math>).  This is modeled on the identity <math>\frac{n}{1}=n</math>.

The field of fractions of <math>R</math> is characterized by the following [[universal property]]:

:if <math>h: R \to F</math> is an [[injective]] [[ring homomorphism]] from <math>R</math> into a field <math>F</math>, then there exists a unique ring homomorphism <math>g: \operatorname{Frac}(R) \to F</math> that extends <math>h</math>.

There is a [[category theory|categorical]] interpretation of this construction.  Let <math>\mathbf{C}</math> be the [[category (mathematics)|category]] of integral domains and injective ring maps. The [[functor]] from <math>\mathbf{C}</math> to the [[category of fields]] that takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the [[adjoint functor|left adjoint]] of the [[inclusion functor]] from the category of fields to <math>\mathbf{C}</math>. Thus the category of fields (which is a full subcategory) is a [[reflective subcategory]] of <math>\mathbf{C}</math>.

A [[multiplicative identity]] is not required for the role of the integral domain; this construction can be applied to any [[zero ring|nonzero]] commutative [[rng (algebra)|rng]] <math>R</math> with no nonzero [[zero divisor]]s. The embedding is given by <math>r\mapsto\frac{rs}{s}</math> for any nonzero <math>s\in R</math>.<ref>{{cite book|last1=Hungerford|first1=Thomas W.|title=Algebra|date=1980|publisher=Springer|location=New York|isbn=3540905189|pages=142–144|edition= Revised 3rd}}</ref>