Field of fractions

<h2 id="definition">Definition</h2>
<p>Given an integral domain 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and letting 
  
    
      
        
          R
          
            ∗
          
        
        =
        R
        ∖
        {
        0
        }
      
    
    {\displaystyle R^{*}=R\setminus \{0\}}
  
, we define an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> on 
  
    
      
        R
        ×
        
          R
          
            ∗
          
        
      
    
    {\displaystyle R\times R^{*}}
  
 by letting 
  
    
      
        (
        n
        ,
        d
        )
        ∼
        (
        m
        ,
        b
        )
      
    
    {\displaystyle (n,d)\sim (m,b)}
  
 whenever 
  
    
      
        n
        b
        =
        m
        d
      
    
    {\displaystyle nb=md}
  
. We denote the <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> of 
  
    
      
        (
        n
        ,
        d
        )
      
    
    {\displaystyle (n,d)}
  
 by 
  
    
      
        
          
            n
            d
          
        
      
    
    {\displaystyle {\frac {n}{d}}}
  
. This notion of equivalence is motivated by the rational numbers 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
, which have the same property with respect to the underlying <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> 
  
    
      
        
          Z
        
      
    
    {\displaystyle \mathbb {Z} }
  
 of integers.
</p><p>Then the field of fractions is the set 
  
    
      
        
          Frac
        
        (
        R
        )
        =
        (
        R
        ×
        
          R
          
            ∗
          
        
        )
        
          /
        
        ∼
      
    
    {\displaystyle {\text{Frac}}(R)=(R\times R^{*})/\sim }
  
 with addition given by
</p>

n
            d
          
        
        +
        
          
            m
            b
          
        
        =
        
          
            
              n
              b
              +
              m
              d
            
            
              d
              b
            
          
        
      
    
    {\displaystyle {\frac {n}{d}}+{\frac {m}{b}}={\frac {nb+md}{db}}}

<p>and multiplication given by 
</p>

n
            d
          
        
        ⋅
        
          
            m
            b
          
        
        =
        
          
            
              n
              m
            
            
              d
              b
            
          
        
        .
      
    
    {\displaystyle {\frac {n}{d}}\cdot {\frac {m}{b}}={\frac {nm}{db}}.}

<p>One may check that these operations are well-defined and that, for any integral domain 
  
    
      
        R
      
    
    {\displaystyle R}
  
, 
  
    
      
        
          Frac
        
        (
        R
        )
      
    
    {\displaystyle {\text{Frac}}(R)}
  
 is indeed a field. In particular, for 
  
    
      
        n
        ,
        d
        ≠
        0
      
    
    {\displaystyle n,d\neq 0}
  
, the multiplicative inverse of 
  
    
      
        
          
            n
            d
          
        
      
    
    {\displaystyle {\frac {n}{d}}}
  
 is as expected: 
  
    
      
        
          
            d
            n
          
        
        ⋅
        
          
            n
            d
          
        
        =
        1
      
    
    {\displaystyle {\frac {d}{n}}\cdot {\frac {n}{d}}=1}
  
.
</p><p>The embedding of 
  
    
      
        R
      
    
    {\displaystyle R}
  
 in 
  
    
      
        Frac
        ⁡
        (
        R
        )
      
    
    {\displaystyle \operatorname {Frac} (R)}
  
 maps each 
  
    
      
        n
      
    
    {\displaystyle n}
  
 in 
  
    
      
        R
      
    
    {\displaystyle R}
  
 to the fraction 
  
    
      
        
          
            
              e
              n
            
            e
          
        
      
    
    {\displaystyle {\frac {en}{e}}}
  
 for any nonzero 
  
    
      
        e
        ∈
        R
      
    
    {\displaystyle e\in R}
  
 (the equivalence class is independent of the choice 
  
    
      
        e
      
    
    {\displaystyle e}
  
).  This is modeled on the identity 
  
    
      
        
          
            n
            1
          
        
        =
        n
      
    
    {\displaystyle {\frac {n}{1}}=n}
  
.
</p><p>The field of fractions of 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is characterized by the following <a href="/facts/Universal_property/5XJBYMNz">universal property</a>:
</p>
if 
  
    
      
        h
        :
        R
        →
        F
      
    
    {\displaystyle h:R\to F}
  
 is an <a href="/facts/Injective/5ES6tqf5">injective</a> <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> from 
  
    
      
        R
      
    
    {\displaystyle R}
  
 into a field 
  
    
      
        F
      
    
    {\displaystyle F}
  
, then there exists a unique ring homomorphism 
  
    
      
        g
        :
        Frac
        ⁡
        (
        R
        )
        →
        F
      
    
    {\displaystyle g:\operatorname {Frac} (R)\to F}
  
 that extends 
  
    
      
        h
      
    
    {\displaystyle h}
  
.
<p>There is a <a href="/facts/Category_theory/6zS3DXPa">categorical</a> interpretation of this construction.  Let 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
 be the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of integral domains and injective ring maps. The <a href="/facts/Functor/l2u3rIBE">functor</a> from 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
 to the <a href="/facts/Category_of_fields/pIwY32bc">category of fields</a> 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 <a href="/facts/Adjoint_functor/bJ1P7AN2">left adjoint</a> of the <a href="/facts/Inclusion_functor/k4lEeDDb">inclusion functor</a> from the category of fields to 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
. Thus the category of fields (which is a full subcategory) is a <a href="/facts/Reflective_subcategory/vaF3ma2y">reflective subcategory</a> of 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
.
</p><p>A <a href="/facts/Multiplicative_identity/9nss3xHY">multiplicative identity</a> is not required for the role of the integral domain; this construction can be applied to any <a href="/facts/Zero_ring/trFjtDJ1">nonzero</a> commutative <a href="/facts/Rng_(algebra)/SquNCXKe">rng</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
 with no nonzero <a href="/facts/Zero_divisor/5GraAU8o">zero divisors</a>. The embedding is given by 
  
    
      
        r
        ↦
        
          
            
              r
              s
            
            s
          
        
      
    
    {\displaystyle r\mapsto {\frac {rs}{s}}}
  
 for any nonzero 
  
    
      
        s
        ∈
        R
      
    
    {\displaystyle s\in R}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="examples">Examples</h2>
<ul><li>The field of fractions of the ring of <a href="/facts/Integer/PUwwrawx">integers</a> is the field of <a href="/facts/Rational_number/VJQmyY05">rationals</a>: 
  
    
      
        
          Q
        
        =
        Frac
        ⁡
        (
        
          Z
        
        )
      
    
    {\displaystyle \mathbb {Q} =\operatorname {Frac} (\mathbb {Z} )}
  
.</li>
<li>Let 
  
    
      
        R
        :=
        {
        a
        +
        b
        
          i
        
        ∣
        a
        ,
        b
        ∈
        
          Z
        
        }
      
    
    {\displaystyle R:=\{a+b\mathrm {i} \mid a,b\in \mathbb {Z} \}}
  
 be the ring of <a href="/facts/Gaussian_integer/zj82AhtL">Gaussian integers</a>. Then 
  
    
      
        Frac
        ⁡
        (
        R
        )
        =
        {
        c
        +
        d
        
          i
        
        ∣
        c
        ,
        d
        ∈
        
          Q
        
        }
      
    
    {\displaystyle \operatorname {Frac} (R)=\{c+d\mathrm {i} \mid c,d\in \mathbb {Q} \}}
  
, the field of <a href="/facts/Gaussian_rational/X0jN1MnI">Gaussian rationals</a>.</li>
<li>The field of fractions of a field is canonically <a href="/facts/Isomorphism/pj8KgkxU">isomorphic</a> to the field itself.</li>
<li>Given a field 
  
    
      
        K
      
    
    {\displaystyle K}
  
, the field of fractions of the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> in one indeterminate 
  
    
      
        K
        [
        X
        ]
      
    
    {\displaystyle K[X]}
  
 (which is an integral domain), is called the <i>field of rational functions</i>, <i>field of rational fractions</i>, or <i>field of rational expressions</i><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> and is denoted 
  
    
      
        K
        (
        X
        )
      
    
    {\displaystyle K(X)}
  
.</li>
<li>The field of fractions of the <a href="/facts/Convolution/PVPrdz9J">convolution</a> ring of half-line functions yields a <a href="/facts/Convolution_quotient/EJzQ8GXW"> space of operators</a>, including the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>, <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a>, and <a href="/facts/Integral_operator/AL6u7Wrj">integral operator</a>.  This construction gives an alternate representation of the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> that does not depend explicitly on an integral transform.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li></ul>
<h2 id="generalizations">Generalizations</h2>
<h3>Localization</h3>
<p class="note">Main article: <a href="/facts/Localization_(commutative_algebra)/rjevzBJf">Localization (commutative algebra)</a></p>
<p>For any <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and any <a href="/facts/Multiplicative_set/WH1VI3b3">multiplicative set</a> 
  
    
      
        S
      
    
    {\displaystyle S}
  
 in 
  
    
      
        R
      
    
    {\displaystyle R}
  
, the <a href="/facts/Localization_of_a_ring/rjevzBJf">localization</a> 
  
    
      
        
          S
          
            −
            1
          
        
        R
      
    
    {\displaystyle S^{-1}R}
  
 is the <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> consisting of <a href="/facts/Fraction/ghnQE9B3">fractions</a>
</p>

r
            s
          
        
      
    
    {\displaystyle {\frac {r}{s}}}

<p>with 
  
    
      
        r
        ∈
        R
      
    
    {\displaystyle r\in R}
  
 and 
  
    
      
        s
        ∈
        S
      
    
    {\displaystyle s\in S}
  
, where now 
  
    
      
        (
        r
        ,
        s
        )
      
    
    {\displaystyle (r,s)}
  
 is equivalent to 
  
    
      
        (
        
          r
          ′
        
        ,
        
          s
          ′
        
        )
      
    
    {\displaystyle (r',s')}
  
 if and only if there exists 
  
    
      
        t
        ∈
        S
      
    
    {\displaystyle t\in S}
  
 such that 
  
    
      
        t
        (
        r
        
          s
          ′
        
        −
        
          r
          ′
        
        s
        )
        =
        0
      
    
    {\displaystyle t(rs'-r's)=0}
  
.
</p><p>Two special cases of this are notable:
</p>
<ul><li>If 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is the complement of a <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a> 
  
    
      
        P
      
    
    {\displaystyle P}
  
, then 
  
    
      
        
          S
          
            −
            1
          
        
        R
      
    
    {\displaystyle S^{-1}R}
  
 is also denoted 
  
    
      
        
          R
          
            P
          
        
      
    
    {\displaystyle R_{P}}
  
.When 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> and 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is the zero ideal, 
  
    
      
        
          R
          
            P
          
        
      
    
    {\displaystyle R_{P}}
  
 is the field of fractions of 
  
    
      
        R
      
    
    {\displaystyle R}
  
.</li>
<li>If 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is the set of non-<a href="/facts/Zero-divisor/5GraAU8o">zero-divisors</a> in 
  
    
      
        R
      
    
    {\displaystyle R}
  
, then 
  
    
      
        
          S
          
            −
            1
          
        
        R
      
    
    {\displaystyle S^{-1}R}
  
 is called the <a href="/facts/Total_quotient_ring/cERmzsNv">total quotient ring</a>.The <a href="/facts/Total_quotient_ring/cERmzsNv">total quotient ring</a> of an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> is its field of fractions, but the <a href="/facts/Total_quotient_ring/cERmzsNv">total quotient ring</a> is defined for any <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>.</li></ul>
<p>Note that it is permitted for 
  
    
      
        S
      
    
    {\displaystyle S}
  
 to contain 0, but in that case 
  
    
      
        
          S
          
            −
            1
          
        
        R
      
    
    {\displaystyle S^{-1}R}
  
 will be the <a href="/facts/Trivial_ring/trFjtDJ1">trivial ring</a>.
</p>
<h3>Semifield of fractions</h3>
<p>The semifield of fractions of a <a href="/facts/Commutative_semiring/7cSvWXVQ">commutative semiring</a> in which every nonzero element is (multiplicatively) cancellative is the smallest <a href="/facts/Semifield/4UsQTJMm">semifield</a> in which it can be <a href="/facts/Embedding/HKeVKc42">embedded</a>. (Note that, unlike the case of rings, a semiring with no <a href="/facts/Zero_divisor/5GraAU8o">zero divisors</a> can still have nonzero elements that are not cancellative. For example, let 
  
    
      
        
          T
        
      
    
    {\displaystyle \mathbb {T} }
  
 denote the <a href="/facts/Tropical_semiring/wEJfp6s1">tropical semiring</a> and let 
  
    
      
        R
        =
        
          T
        
        [
        X
        ]
      
    
    {\displaystyle R=\mathbb {T} [X]}
  
 be the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial semiring</a> over 
  
    
      
        
          T
        
      
    
    {\displaystyle \mathbb {T} }
  
. Then 
  
    
      
        R
      
    
    {\displaystyle R}
  
 has no zero divisors, but the element 
  
    
      
        1
        +
        X
      
    
    {\displaystyle 1+X}
  
 is not cancellative because 
  
    
      
        (
        1
        +
        X
        )
        (
        1
        +
        X
        +
        
          X
          
            2
          
        
        )
        =
        1
        +
        X
        +
        
          X
          
            2
          
        
        +
        
          X
          
            3
          
        
        =
        (
        1
        +
        X
        )
        (
        1
        +
        
          X
          
            2
          
        
        )
      
    
    {\displaystyle (1+X)(1+X+X^{2})=1+X+X^{2}+X^{3}=(1+X)(1+X^{2})}
  
).
</p><p>The elements of the semifield of fractions of the commutative <a href="/facts/Semiring/7cSvWXVQ">semiring</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
 are <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a> written as
</p>

a
            b
          
        
      
    
    {\displaystyle {\frac {a}{b}}}

<p>with 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 in 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and 
  
    
      
        b
        ≠
        0
      
    
    {\displaystyle b\neq 0}
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Ore_condition/FXaaof55">Ore condition</a>; condition related to constructing fractions in the noncommutative case.</li>
<li><a href="/facts/Total_ring_of_fractions/cERmzsNv">Total ring of fractions</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Hungerford, Thomas W. (1980). Algebra (Revised 3rd ed.). New York: Springer. pp. 142–144. ISBN 3540905189. <a href="3540905189" target="_blank">3540905189</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Vinberg, Ėrnest Borisovich (2003). A course in algebra. American Mathematical Society. p. 131. ISBN 978-0-8218-8394-5. <a href="978-0-8218-8394-5" target="_blank">978-0-8218-8394-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Foldes, Stephan (1994). Fundamental structures of algebra and discrete mathematics. Wiley. p. 128. ISBN 0-471-57180-6. <a href="0-471-57180-6" target="_blank">0-471-57180-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Grillet, Pierre Antoine (2007). "3.5 Rings: Polynomials in One Variable". Abstract algebra. Springer. p. 124. ISBN 978-0-387-71568-1. <a href="978-0-387-71568-1" target="_blank">978-0-387-71568-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Marecek, Lynn; Mathis, Andrea Honeycutt (6 May 2020). Intermediate Algebra 2e. OpenStax. §7.1. <a href="https://openstax.org/details/books/intermediate-algebra-2e" target="_blank">https://openstax.org/details/books/intermediate-algebra-2e</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Mikusiński, Jan (14 July 2014). Operational Calculus. Elsevier. ISBN 9781483278933. <a href="9781483278933" target="_blank">9781483278933</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Field of fractions open-in-new

Field of fractions