Torsionless module

<h2 id="properties-and-examples">Properties and examples</h2>
A module is torsionless if and only if the canonical map into its double <a href="/facts/Dual_module/Q89wXgxc">dual</a>,

M
        →
        
          M
          
            ∗
            ∗
          
        
        =
        
          Hom
          
            R
          
        
        ⁡
        (
        
          M
          
            ∗
          
        
        ,
        R
        )
        ,
        
        m
        ↦
        (
        f
        ↦
        f
        (
        m
        )
        )
        ,
        m
        ∈
        M
        ,
        f
        ∈
        
          M
          
            ∗
          
        
        ,
      
    
    {\displaystyle M\to M^{\ast \ast }=\operatorname {Hom} _{R}(M^{\ast },R),\quad m\mapsto (f\mapsto f(m)),m\in M,f\in M^{\ast },}

is <a href="/facts/Injective/5ES6tqf5">injective</a>. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive.

<ul><li>A unital <a href="/facts/Free_module/o6rC6yoJ">free module</a> is torsionless. More generally, a <a href="/facts/Direct_sum/x3ljVsP6">direct sum</a> of torsionless modules is torsionless.</li>
<li>A free module is reflexive if it is <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a>, and for some rings there are also infinitely generated free modules that are reflexive. For instance, the direct sum of countably many copies of the integers is a reflexive module over the integers, see for instance.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li></ul>
<ul><li>A submodule of a torsionless module is torsionless. In particular, any <a href="/facts/Projective_module/lHymqhNC">projective module</a> over R is torsionless; any left ideal of R is a torsionless left module, and similarly for the right ideals.</li>
<li>Any torsionless module over a <a href="/facts/Domain_(ring_theory)/p4eKRC8W">domain</a> is a <a href="/facts/Torsion-free_module/IFsk1xw0">torsion-free module</a>, but the converse is not true, as Q is a torsion-free Z-module that is not torsionless.</li>
<li>If R is a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> that is an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> and M is a <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> torsion-free module then M can be embedded into Rn, and hence M is torsionless.</li>
<li>Suppose that N is a right R-module, then its dual N∗ has a structure of a left R-module. It turns out that any left R-module arising in this way is torsionless (similarly, any right R-module that is a dual of a left R-module is torsionless).</li>
<li>Over a <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domain</a>, a finitely generated module is reflexive if and only if it is torsion-free.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>Let R be a <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a> and M a reflexive finitely generated module over R. Then 
 
 
 
 M
 
 ⊗
 
 R
 
 
 S
 
 
 {\displaystyle M\otimes _{R}S}
 
 is a reflexive module over S whenever S is <a href="/facts/Flat_module/1WFrnJP3">flat</a> over R.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li></ul>
<h2 id="relation-with-semihereditary-rings">Relation with semihereditary rings</h2>
Stephen Chase proved the following characterization of <a href="/facts/Semihereditary_ring/w2yUaqFe">semihereditary rings</a> in connection with torsionless modules:
For any ring R, the following conditions are equivalent:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<ul><li>R is left semihereditary.</li>
<li>All torsionless right R-modules are <a href="/facts/Flat_module/1WFrnJP3">flat</a>.</li>
<li>The ring R is left <a href="/facts/Coherent_ring/JV8ApsMo">coherent</a> and satisfies any of the four conditions that are known to be equivalent:
<ul><li>All right ideals of R are flat.</li>
<li>All left ideals of R are flat.</li>
<li>Submodules of all right flat R-modules are flat.</li>
<li>Submodules of all left flat R-modules are flat.</li></ul></li></ul>
(The mixture of left/right adjectives in the statement is not a mistake.)

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Pr%25C3%25BCfer_domain/YQtQooMq">Prüfer domain</a></li>
<li><a href="/facts/Reflexive_sheaf/IvoIt2Gh">reflexive sheaf</a></li></ul>
<h2 id="note">Note</h2>

<ul><li>Chapter VII of <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1998), Commutative algebra (2nd ed.), <a href="/facts/Springer_Verlag/nAesf6nT">Springer Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-64239-0</li>
<li><a href="/facts/Tsit_Yuen_Lam/S1ewInkE">Lam, Tsit Yuen</a> (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">Eklof, P. C.; Mekler, A. H. (2002). Almost Free Modules - Set-theoretic Methods. North-Holland Mathematical Library. Vol. 65. doi:10.1016/s0924-6509(02)x8001-5. ISBN 9780444504920. S2CID 116961421. <a href="9780444504920" target="_blank">9780444504920</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Proof: If M is reflexive, it is torsionless, thus is a submodule of a finitely generated projective module and hence is projective (semi-hereditary condition). Conversely, over a Dedekind domain, a finitely generated torsion-free module is projective and a projective module is reflexive (the existence of a dual basis). <a href="/wiki/Dual_basis" target="_blank">/wiki/Dual_basis</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Bourbaki 1998, p. Ch. VII, § 4, n. 2. Proposition 8. - Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-540-64239-0 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lam 1999, p 146. - 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:4" class="footnote-back-ref">↩</a></li>
</ol>

Torsionless module open-in-new

Torsionless module