Quotient ring

<p>In <a href="/facts/Ring_theory/VW6svEFe">ring theory</a>, a branch of <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> in <a href="/facts/Group_theory/NfowcI5H">group theory</a> and to the <a href="/facts/Quotient_space_(linear_algebra)/WmQ3lqdI">quotient space</a> in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>. It is a specific example of a <a href="/facts/Quotient_(universal_algebra)/8uQzYkm7">quotient</a>, as viewed from the general setting of <a href="/facts/Universal_algebra/uMJAL6ER">universal algebra</a>. Starting with a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and a <a href="/facts/Two-sided_ideal/vCvytduX">two-sided ideal</a> 
  
    
      
        I
      
    
    {\displaystyle I}
  
 in ⁠
  
    
      
        R
      
    
    {\displaystyle R}
  
⁠, a new ring, the quotient ring ⁠
  
    
      
        R
         
        
          /
        
         
        I
      
    
    {\displaystyle R\ /\ I}
  
⁠, is constructed, whose elements are the <a href="/facts/Cosets/zFgQUTLY">cosets</a> of 
  
    
      
        I
      
    
    {\displaystyle I}
  
 in 
  
    
      
        R
      
    
    {\displaystyle R}
  
 subject to special 
  
    
      
        +
      
    
    {\displaystyle +}
  
 and 
  
    
      
        ⋅
      
    
    {\displaystyle \cdot }
  
 operations. (Quotient ring notation always uses a <a href="/facts/Fraction_slash/fuaRyKzc">fraction slash</a> "⁠
  
    
      
        
          /
        
      
    
    {\displaystyle /}
  
⁠".)
</p><p>Quotient rings are distinct from the so-called "quotient field", or <a href="/facts/Field_of_fractions/cmlJrDSS">field of fractions</a>, of an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> as well as from the more general "rings of quotients" obtained by <a href="/facts/Localization_of_a_ring/rjevzBJf">localization</a>.
</p>

Quotient ring open-in-new

Quotient ring