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Math Help - Isomorphism on Set of Homomorphisms Question (Rotman Advanced Algebra)

  1. #1
    Newbie MaximalIdeal's Avatar
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    Isomorphism on Set of Homomorphisms Question (Rotman Advanced Algebra)

    Hi, hope you can help me with a question that has been annoying me.

    It's from Rotman Advanced Modern Algebra, Ex. 7.5.

    For every R-module M, prove that there is an R-isomorphism

    \varphi_{m}:Hom_{R}(R,M)\rightarrow\mbox{M}

    given by

    \varphi_{m}:\mbox{f}\mapsto\mbox{f(1)}

    I have got everything apart from the part of the proof that \varphi_{m} is surjective.

    I've been turning it over and I think I'm just missing something. Any ideas? Any help is much appreciated. I'm new here, am starting postgrad maths soon and look forward to getting to know you guys.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by MaximalIdeal View Post
    I have got everything apart from the part of the proof that \varphi_{m} is surjective.

    Given x\in M, define:

    f_x:R\rightarrow M\;,\quad f_x(\lambda)=\lambda x


    Fernando Revilla
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  3. #3
    Newbie MaximalIdeal's Avatar
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    Great, makes perfect sense, thanks!
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by MaximalIdeal View Post
    Great, makes perfect sense, thanks!

    You are welcome


    Fernando Revilla
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