Isomorphism on Set of Homomorphisms Question (Rotman Advanced Algebra)

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January 27th 2011, 01:54 PM
MaximalIdeal

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.
January 27th 2011, 03:00 PM
FernandoRevilla

Quote:

Originally Posted by MaximalIdeal

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
January 27th 2011, 03:13 PM
MaximalIdeal

Great, makes perfect sense, thanks!
January 27th 2011, 03:15 PM
FernandoRevilla

Quote:

Originally Posted by MaximalIdeal

Great, makes perfect sense, thanks!

You are welcome :)

Fernando Revilla

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