Flat Modules are Torsion-free












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Can anyone help me prove that a flat module is torsion-free? I’ve tried supposing is isn’t and then get the contradiction that it isn’t flat, but I’m not having any good results. Any suggestion is welcomed. Thanks.






abstract-algebra modules flatness







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  • $begingroup$
    Consider, for a regular element $rin R$, the exact sequence $0to Rto Rto R/(r)to 0$
    $endgroup$
    – Max
    Jan 27 at 17:55










  • $begingroup$
    Ok yeah , that is going to be a s.e.s, i thought about doing something similiar but i dont see where i get what i want.
    $endgroup$
    – Pedro Santos
    Jan 27 at 18:52










  • $begingroup$
    Well you have a flat module and a short exact sequence, what can you do with it ?
    $endgroup$
    – Max
    Jan 27 at 19:10










  • $begingroup$
    Get another short exat sequence
    $endgroup$
    – Pedro Santos
    Jan 27 at 19:13
















0












$begingroup$


Can anyone help me prove that a flat module is torsion-free? I’ve tried supposing is isn’t and then get the contradiction that it isn’t flat, but I’m not having any good results. Any suggestion is welcomed. Thanks.






abstract-algebra modules flatness







share|cite|improve this question











$endgroup$












  • $begingroup$
    Consider, for a regular element $rin R$, the exact sequence $0to Rto Rto R/(r)to 0$
    $endgroup$
    – Max
    Jan 27 at 17:55










  • $begingroup$
    Ok yeah , that is going to be a s.e.s, i thought about doing something similiar but i dont see where i get what i want.
    $endgroup$
    – Pedro Santos
    Jan 27 at 18:52










  • $begingroup$
    Well you have a flat module and a short exact sequence, what can you do with it ?
    $endgroup$
    – Max
    Jan 27 at 19:10










  • $begingroup$
    Get another short exat sequence
    $endgroup$
    – Pedro Santos
    Jan 27 at 19:13














0












0








0





$begingroup$


Can anyone help me prove that a flat module is torsion-free? I’ve tried supposing is isn’t and then get the contradiction that it isn’t flat, but I’m not having any good results. Any suggestion is welcomed. Thanks.






abstract-algebra modules flatness







share|cite|improve this question











$endgroup$




Can anyone help me prove that a flat module is torsion-free? I’ve tried supposing is isn’t and then get the contradiction that it isn’t flat, but I’m not having any good results. Any suggestion is welcomed. Thanks.






abstract-algebra modules flatness




abstract-algebra modules flatness






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 27 at 17:37









J. W. Tanner

3,8001320




3,8001320










asked Jan 27 at 16:41









Pedro SantosPedro Santos

1609




1609












  • $begingroup$
    Consider, for a regular element $rin R$, the exact sequence $0to Rto Rto R/(r)to 0$
    $endgroup$
    – Max
    Jan 27 at 17:55










  • $begingroup$
    Ok yeah , that is going to be a s.e.s, i thought about doing something similiar but i dont see where i get what i want.
    $endgroup$
    – Pedro Santos
    Jan 27 at 18:52










  • $begingroup$
    Well you have a flat module and a short exact sequence, what can you do with it ?
    $endgroup$
    – Max
    Jan 27 at 19:10










  • $begingroup$
    Get another short exat sequence
    $endgroup$
    – Pedro Santos
    Jan 27 at 19:13


















  • $begingroup$
    Consider, for a regular element $rin R$, the exact sequence $0to Rto Rto R/(r)to 0$
    $endgroup$
    – Max
    Jan 27 at 17:55










  • $begingroup$
    Ok yeah , that is going to be a s.e.s, i thought about doing something similiar but i dont see where i get what i want.
    $endgroup$
    – Pedro Santos
    Jan 27 at 18:52










  • $begingroup$
    Well you have a flat module and a short exact sequence, what can you do with it ?
    $endgroup$
    – Max
    Jan 27 at 19:10










  • $begingroup$
    Get another short exat sequence
    $endgroup$
    – Pedro Santos
    Jan 27 at 19:13
















$begingroup$
Consider, for a regular element $rin R$, the exact sequence $0to Rto Rto R/(r)to 0$
$endgroup$
– Max
Jan 27 at 17:55




$begingroup$
Consider, for a regular element $rin R$, the exact sequence $0to Rto Rto R/(r)to 0$
$endgroup$
– Max
Jan 27 at 17:55












$begingroup$
Ok yeah , that is going to be a s.e.s, i thought about doing something similiar but i dont see where i get what i want.
$endgroup$
– Pedro Santos
Jan 27 at 18:52




$begingroup$
Ok yeah , that is going to be a s.e.s, i thought about doing something similiar but i dont see where i get what i want.
$endgroup$
– Pedro Santos
Jan 27 at 18:52












$begingroup$
Well you have a flat module and a short exact sequence, what can you do with it ?
$endgroup$
– Max
Jan 27 at 19:10




$begingroup$
Well you have a flat module and a short exact sequence, what can you do with it ?
$endgroup$
– Max
Jan 27 at 19:10












$begingroup$
Get another short exat sequence
$endgroup$
– Pedro Santos
Jan 27 at 19:13




$begingroup$
Get another short exat sequence
$endgroup$
– Pedro Santos
Jan 27 at 19:13










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