Topological module

<h2 id="examples">Examples</h2>
<p>A <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> is a topological module over a <a href="/facts/Topological_field/qxQqzhCy">topological field</a>.
</p><p>An <a href="/facts/Abelian_group/FfWwmx4R">abelian</a> <a href="/facts/Topological_group/cAgNMMRU">topological group</a> can be considered as a topological module over 
  
    
      
        
          Z
        
        ,
      
    
    {\displaystyle \mathbb {Z} ,}
  
 where 
  
    
      
        
          Z
        
      
    
    {\displaystyle \mathbb {Z} }
  
 is the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> with the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a>.
</p><p>A topological ring is a topological module over each of its <a href="/facts/Subring/M0perybt">subrings</a>.
</p><p>A more complicated example is the 
  
    
      
        I
      
    
    {\displaystyle I}
  
-<a href="/facts/Adic_topology/ALsc1Tc6">adic topology</a> on a ring and its modules. Let 
  
    
      
        I
      
    
    {\displaystyle I}
  
 be an <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> of a ring 
  
    
      
        R
        .
      
    
    {\displaystyle R.}
  
 The sets of the form 
  
    
      
        x
        +
        
          I
          
            n
          
        
      
    
    {\displaystyle x+I^{n}}
  
 for all 
  
    
      
        x
        ∈
        R
      
    
    {\displaystyle x\in R}
  
 and all positive integers 
  
    
      
        n
        ,
      
    
    {\displaystyle n,}
  
 form a <a href="/facts/Base_(topology)/8EDSDVPD">base</a> for a topology on 
  
    
      
        R
      
    
    {\displaystyle R}
  
 that makes 
  
    
      
        R
      
    
    {\displaystyle R}
  
 into a topological ring. Then for any left 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module 
  
    
      
        M
        ,
      
    
    {\displaystyle M,}
  
 the sets of the form 
  
    
      
        x
        +
        
          I
          
            n
          
        
        M
        ,
      
    
    {\displaystyle x+I^{n}M,}
  
 for all 
  
    
      
        x
        ∈
        M
      
    
    {\displaystyle x\in M}
  
 and all positive integers 
  
    
      
        n
        ,
      
    
    {\displaystyle n,}
  
 form a base for a topology on 
  
    
      
        M
      
    
    {\displaystyle M}
  
 that makes 
  
    
      
        M
      
    
    {\displaystyle M}
  
 into a topological module over the topological ring 
  
    
      
        R
        .
      
    
    {\displaystyle R.}

</p>
<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Linear_topology/5w5TpKx9">Linear topology</a></li>
<li><a href="/facts/Ordered_topological_vector_space/N1fvN8th">Ordered topological vector space</a></li>
<li><a href="/facts/Topological_abelian_group/u8FSbRux">Topological abelian group</a> – topological group whose group is abelianPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Topological_field/qxQqzhCy">Topological field</a> – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Topological_group/cAgNMMRU">Topological group</a> – Group that is a topological space with continuous group action</li>
<li><a href="/facts/Topological_ring/qxQqzhCy">Topological ring</a> – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Topological_semigroup/JDwmyd5M">Topological semigroup</a> – semigroup with continuous operationPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Topological_vector_space/XRex1ChN">Topological vector space</a> – Vector space with a notion of nearness</li></ul>

<ul><li><a href="/facts/Michael_Atiyah/RQVJ9ATE">Atiyah, Michael Francis</a>; <a href="/facts/Ian_G._Macdonald/PoaeNbeg">MacDonald, I.G.</a> (1969). <i>Introduction to Commutative Algebra</i>. Westview Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-40751-8.</li>
<li>Kuz'min, L. V. (1993). "Topological modules". In Hazewinkel, M. (ed.). <i>Encyclopedia of Mathematics</i>. Vol. 9. <a href="/facts/Kluwer_Academic_Publishers/nAesf6nT">Kluwer Academic Publishers</a>.</li></ul>

Topological module open-in-new

Topological module