Pure submodule

<h2 id="definition">Definition</h2>
Let R be a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> (associative, with 1), let M be a (left) <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over R, let P be a <a href="/facts/Submodule/jP0LIhFL">submodule</a> of M and let i: P → M be the natural <a href="/facts/Injective/5ES6tqf5">injective</a> map. Then P is a pure submodule of M if, for any (right) R-module X, the natural induced map idX ⊗ i : X ⊗ P → X ⊗ M (where the <a href="/facts/Tensor_product/7K3pR1ap">tensor products</a> are taken over R) is injective.
Analogously, a <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequence</a>

0
        ⟶
        A
        
         
        
          
            
              
                ⟶
              
              
                f
              
            
          
        
         
        B
        
         
        
          
            
              
                ⟶
              
              
                g
              
            
          
        
         
        C
        ⟶
        0
      
    
    {\displaystyle 0\longrightarrow A\,\ {\stackrel {f}{\longrightarrow }}\ B\,\ {\stackrel {g}{\longrightarrow }}\ C\longrightarrow 0}

of (left) R-modules is pure exact if the sequence stays exact when tensored with any (right) R-module X. This is equivalent to saying that f(A) is a pure submodule of B.

<h2 id="equivalent-characterizations">Equivalent characterizations</h2>
Purity of a submodule can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations. Specifically, P is pure in M if and only if the following condition holds: for any m-by-n <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> (aij) with entries in R, and any set y1, ..., ym of elements of P, if there exist elements x1, ..., xn in M such that

∑
          
            j
            =
            1
          
          
            n
          
        
        
          a
          
            i
            j
          
        
        
          x
          
            j
          
        
        =
        
          y
          
            i
          
        
        
        
          
             for 
          
        
        i
        =
        1
        ,
        …
        ,
        m
      
    
    {\displaystyle \sum _{j=1}^{n}a_{ij}x_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m}

then there also exist elements x1′, ..., xn′ in P such that

∑
          
            j
            =
            1
          
          
            n
          
        
        
          a
          
            i
            j
          
        
        
          x
          
            j
          
          ′
        
        =
        
          y
          
            i
          
        
        
        
          
             for 
          
        
        i
        =
        1
        ,
        …
        ,
        m
      
    
    {\displaystyle \sum _{j=1}^{n}a_{ij}x'_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m}

Another characterization is: a sequence is pure exact if and only if it is the <a href="/facts/Filtered_colimit/dxdcQtJQ">filtered colimit</a> (also known as <a href="/facts/Direct_limit/V3JuHnYK">direct limit</a>) of <a href="/facts/Split_exact_sequence/fBswKvU9">split exact sequences</a>

0
 ⟶
 
 A
 
 i
 
 
 ⟶
 
 B
 
 i
 
 
 ⟶
 
 C
 
 i
 
 
 ⟶
 0.
 
 
 {\displaystyle 0\longrightarrow A_{i}\longrightarrow B_{i}\longrightarrow C_{i}\longrightarrow 0.}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
<h2 id="examples">Examples</h2>
<ul><li>Every <a href="/facts/Direct_summand/x3ljVsP6">direct summand</a> of M is pure in M. Consequently, every <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> is pure.</li></ul>
<h2 id="properties">Properties</h2>
Suppose<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

is a short exact sequence of R-modules, then: 

<ol><li>C is a <a href="/facts/Flat_module/1WFrnJP3">flat module</a> if and only if the exact sequence is pure exact for every A and B. From this we can deduce that over a <a href="/facts/Von_Neumann_regular_ring/RMVqKKkk">von Neumann regular ring</a>, every submodule of every R-module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>Suppose B is flat. Then the sequence is pure exact if and only if C is flat. From this one can deduce that pure submodules of flat modules are flat.</li>
<li>Suppose C is flat. Then B is flat if and only if A is flat.</li></ol>

If 
 
 
 
 0
 ⟶
 A
 
  
 
 
 
 
 ⟶
 
 
 f
 
 
 
 
  
 B
 
  
 
 
 
 
 ⟶
 
 
 g
 
 
 
 
  
 C
 ⟶
 0
 
 
 {\displaystyle 0\longrightarrow A\,\ {\stackrel {f}{\longrightarrow }}\ B\,\ {\stackrel {g}{\longrightarrow }}\ C\longrightarrow 0}
 
 is pure-exact, and F is a <a href="/facts/Finitely_presented_module/6Njcpgzr">finitely presented</a> R-module, then every homomorphism from F to C can be lifted to B, i.e. to every u : F → C there exists v : F → B such that gv=u.

<ul><li>Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783319194226</li></ul>
<ul><li>Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98428-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1653294">1653294</a></li></ul>

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

<ol>
<li id="fn:1">For abelian groups, this is proved in Fuchs (2015, Ch. 5, Thm. 3.4) - Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN 9783319194226 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lam 1999, p. 154. - Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1653294" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1653294</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Lam 1999, p. 162. - Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1653294" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1653294</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Pure submodule open-in-new

Pure submodule