Isomorphism on Set of Homomorphisms Question (Rotman Advanced Algebra)

• 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