Field of fractions

<p>In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, the field of fractions of an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> is the smallest <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> in which it can be <a href="/facts/Embedding/HKeVKc42">embedded</a>. The construction of the field of fractions is modeled on the relationship between the integral domain of <a href="/facts/Integer/PUwwrawx">integers</a> and the field of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>. Intuitively, it consists of ratios between integral domain elements.
</p><p>The field of fractions of an integral domain 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is sometimes denoted by 
  
    
      
        Frac
        ⁡
        (
        R
        )
      
    
    {\displaystyle \operatorname {Frac} (R)}
  
 or 
  
    
      
        Quot
        ⁡
        (
        R
        )
      
    
    {\displaystyle \operatorname {Quot} (R)}
  
, and the construction is sometimes also called the fraction field, field of quotients, or quotient field of 
  
    
      
        R
      
    
    {\displaystyle R}
  
.  All four are in common usage, but are not to be confused with the <a href="/facts/Quotient_ring/X2WkXyEw">quotient of a ring by an ideal</a>, which is a quite different concept. For a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> that is not an integral domain, the analogous construction is called the <a href="/facts/Localization_(commutative_algebra)/rjevzBJf">localization</a> or ring of quotients.
</p>

Field of fractions open-in-new

Field of fractions