$M$ is an irreducible $R$ module $iff$ $M$ is a cyclic module and every nonzero element is a generator












2












$begingroup$


$M$ is an irreducible $R$ module $iff$ $M$ is a cyclic module and every nonzero element is a generator.



($rightarrow$) If $M$ is an irreducible $R$-module then it's obvious that $M$ is a cylclic module where every nonzero element is a generator, otherwise $mR <_R M$ would be a submodule of M.



($leftarrow$) So, this would be obvious if we knew that every $M$ submodule is somehow related to the cyclic submodules of $M$, like a direct product or something. If $M$ was free then this would be easy, but as it is this suprises me, I feel like there should be some weird counterexample where there is some module $M$ with some submodule not related to its cyclic submodules. Why is this not possible?










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  • 1




    $begingroup$
    Let $N$ be any nonzero submodule of $M$. Pick any nonzero element $x in N$. $x$ generates $M$, thus $M subseteq N$. End of the proof.
    $endgroup$
    – Crostul
    Dec 9 '18 at 20:45


















2












$begingroup$


$M$ is an irreducible $R$ module $iff$ $M$ is a cyclic module and every nonzero element is a generator.



($rightarrow$) If $M$ is an irreducible $R$-module then it's obvious that $M$ is a cylclic module where every nonzero element is a generator, otherwise $mR <_R M$ would be a submodule of M.



($leftarrow$) So, this would be obvious if we knew that every $M$ submodule is somehow related to the cyclic submodules of $M$, like a direct product or something. If $M$ was free then this would be easy, but as it is this suprises me, I feel like there should be some weird counterexample where there is some module $M$ with some submodule not related to its cyclic submodules. Why is this not possible?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Let $N$ be any nonzero submodule of $M$. Pick any nonzero element $x in N$. $x$ generates $M$, thus $M subseteq N$. End of the proof.
    $endgroup$
    – Crostul
    Dec 9 '18 at 20:45
















2












2








2





$begingroup$


$M$ is an irreducible $R$ module $iff$ $M$ is a cyclic module and every nonzero element is a generator.



($rightarrow$) If $M$ is an irreducible $R$-module then it's obvious that $M$ is a cylclic module where every nonzero element is a generator, otherwise $mR <_R M$ would be a submodule of M.



($leftarrow$) So, this would be obvious if we knew that every $M$ submodule is somehow related to the cyclic submodules of $M$, like a direct product or something. If $M$ was free then this would be easy, but as it is this suprises me, I feel like there should be some weird counterexample where there is some module $M$ with some submodule not related to its cyclic submodules. Why is this not possible?










share|cite|improve this question









$endgroup$




$M$ is an irreducible $R$ module $iff$ $M$ is a cyclic module and every nonzero element is a generator.



($rightarrow$) If $M$ is an irreducible $R$-module then it's obvious that $M$ is a cylclic module where every nonzero element is a generator, otherwise $mR <_R M$ would be a submodule of M.



($leftarrow$) So, this would be obvious if we knew that every $M$ submodule is somehow related to the cyclic submodules of $M$, like a direct product or something. If $M$ was free then this would be easy, but as it is this suprises me, I feel like there should be some weird counterexample where there is some module $M$ with some submodule not related to its cyclic submodules. Why is this not possible?







abstract-algebra






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asked Dec 9 '18 at 20:34









Math is hardMath is hard

817211




817211








  • 1




    $begingroup$
    Let $N$ be any nonzero submodule of $M$. Pick any nonzero element $x in N$. $x$ generates $M$, thus $M subseteq N$. End of the proof.
    $endgroup$
    – Crostul
    Dec 9 '18 at 20:45
















  • 1




    $begingroup$
    Let $N$ be any nonzero submodule of $M$. Pick any nonzero element $x in N$. $x$ generates $M$, thus $M subseteq N$. End of the proof.
    $endgroup$
    – Crostul
    Dec 9 '18 at 20:45










1




1




$begingroup$
Let $N$ be any nonzero submodule of $M$. Pick any nonzero element $x in N$. $x$ generates $M$, thus $M subseteq N$. End of the proof.
$endgroup$
– Crostul
Dec 9 '18 at 20:45






$begingroup$
Let $N$ be any nonzero submodule of $M$. Pick any nonzero element $x in N$. $x$ generates $M$, thus $M subseteq N$. End of the proof.
$endgroup$
– Crostul
Dec 9 '18 at 20:45












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$begingroup$

Note that the statement is not really correct: a module $M$ is irreducible if and only if it is nonzero and every nonzero element of $M$ is a generator.



You're looking too far, in my opinion.



Suppose every nonzero element of $M$ (a nonzero module) is a generator. Let $L$ be a nonzero submodule of $M$; then take $xin L$, $xne0$; then $M=xRsubseteq L$, forcing $L=M$.



For the converse: let $xin M$, $xne0$; then $xR$ is a nonzero submodule of $M$, so $xR=M$ and $x$ is a generator.






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    $begingroup$

    Note that the statement is not really correct: a module $M$ is irreducible if and only if it is nonzero and every nonzero element of $M$ is a generator.



    You're looking too far, in my opinion.



    Suppose every nonzero element of $M$ (a nonzero module) is a generator. Let $L$ be a nonzero submodule of $M$; then take $xin L$, $xne0$; then $M=xRsubseteq L$, forcing $L=M$.



    For the converse: let $xin M$, $xne0$; then $xR$ is a nonzero submodule of $M$, so $xR=M$ and $x$ is a generator.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Note that the statement is not really correct: a module $M$ is irreducible if and only if it is nonzero and every nonzero element of $M$ is a generator.



      You're looking too far, in my opinion.



      Suppose every nonzero element of $M$ (a nonzero module) is a generator. Let $L$ be a nonzero submodule of $M$; then take $xin L$, $xne0$; then $M=xRsubseteq L$, forcing $L=M$.



      For the converse: let $xin M$, $xne0$; then $xR$ is a nonzero submodule of $M$, so $xR=M$ and $x$ is a generator.






      share|cite|improve this answer









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        1












        1








        1





        $begingroup$

        Note that the statement is not really correct: a module $M$ is irreducible if and only if it is nonzero and every nonzero element of $M$ is a generator.



        You're looking too far, in my opinion.



        Suppose every nonzero element of $M$ (a nonzero module) is a generator. Let $L$ be a nonzero submodule of $M$; then take $xin L$, $xne0$; then $M=xRsubseteq L$, forcing $L=M$.



        For the converse: let $xin M$, $xne0$; then $xR$ is a nonzero submodule of $M$, so $xR=M$ and $x$ is a generator.






        share|cite|improve this answer









        $endgroup$



        Note that the statement is not really correct: a module $M$ is irreducible if and only if it is nonzero and every nonzero element of $M$ is a generator.



        You're looking too far, in my opinion.



        Suppose every nonzero element of $M$ (a nonzero module) is a generator. Let $L$ be a nonzero submodule of $M$; then take $xin L$, $xne0$; then $M=xRsubseteq L$, forcing $L=M$.



        For the converse: let $xin M$, $xne0$; then $xR$ is a nonzero submodule of $M$, so $xR=M$ and $x$ is a generator.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 9 '18 at 20:45









        egregegreg

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        181k1485203






























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