Simple ring

<h2 id="examples">Examples</h2>
<p>Let 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 be the field of real numbers, 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
 be the field of complex numbers, and 
  
    
      
        
          H
        
      
    
    {\displaystyle \mathbb {H} }
  
 the <a href="/facts/Quaternion/T3tnu7mX">quaternions</a>.
</p>
<ul><li>A <a href="/facts/Central_simple_algebra/RsBMUkIW">central simple algebra</a> (sometimes called a Brauer algebra) is a simple finite-dimensional algebra over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        F
      
    
    {\displaystyle F}
  
 whose <a href="/facts/Center_of_an_algebra/kJcejgkj">center</a> is 
  
    
      
        F
      
    
    {\displaystyle F}
  
.</li>
<li>Every finite-dimensional simple algebra over 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 is isomorphic to an algebra of 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 matrices with entries in 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
, 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
, or 
  
    
      
        
          H
        
      
    
    {\displaystyle \mathbb {H} }
  
.  Every <a href="/facts/Central_simple_algebra/RsBMUkIW">central simple algebra</a> over 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 is isomorphic to an algebra of 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 matrices with entries 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 or 
  
    
      
        
          H
        
      
    
    {\displaystyle \mathbb {H} }
  
.  These results follow from the <a href="/facts/Frobenius_theorem_(real_division_algebras)/bX6ox4mj">Frobenius theorem</a>.</li>
<li>Every finite-dimensional simple algebra over 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
 is a central simple algebra, and is isomorphic to a matrix ring over 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
.</li>
<li>Every finite-dimensional central simple algebra over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> is isomorphic to a matrix ring over that field.</li>
<li>Over a field of characteristic zero, the <a href="/facts/Weyl_algebra/PADSD0k9">Weyl algebra</a> is simple but not semisimple, and in particular not a matrix algebra over a division algebra over its center; the Weyl algebra is infinite-dimensional, so Wedderburn's theorem does not apply to it.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Simple_(algebra)/dBdUtS9Y">Simple (algebra)</a></li>
<li><a href="/facts/Simple_algebra_(universal_algebra)/tpYhp2Tc">Simple algebra (universal algebra)</a></li></ul>

<ul><li><a href="/facts/Abraham_Adrian_Albert/whzaVMMg">Albert, A. A.</a> (2003). <i>Structure of Algebras</i>. Colloquium publications. Vol. 24. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. p. 37. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1024-3.</li>
<li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (2012), <i>Algèbre Ch. 8</i> (2nd ed.), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-35315-7</li>
<li>Nicholson, William K. (1993). <a href="https://www.thebookshelf.auckland.ac.nz/docs/NZJMaths/nzjmaths022/nzjmaths022-01-010.pdf">"A short proof of the Wedderburn-Artin theorem"</a> (PDF). <i>New Zealand J. Math</i>. 22: 83–86.</li>
<li>Henderson, D. W. (1965). "A short proof of Wedderburn's theorem". <i>Amer. Math. Monthly</i>. 72: 385–386. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2313499">10.2307/2313499</a>.</li>
<li><a href="/facts/Tsit_Yuen_Lam/S1ewInkE">Lam, Tsit-Yuen</a> (2001), <i>A First Course in Noncommutative Rings</i> (2nd ed.), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4419-8616-0">10.1007/978-1-4419-8616-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95325-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1838439">1838439</a></li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (2002), <i>Algebra</i> (3rd ed.), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0387953854</li>
<li><a href="/facts/Nathan_Jacobson/TEDrg4Lk">Jacobson, Nathan</a> (1989), <i>Basic Algebra II</i> (2nd ed.), W. H. Freeman, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7167-1933-5</li></ul>

Simple ring open-in-new

Simple ring