1. 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

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

given by

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

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

2. Originally Posted by MaximalIdeal
I have got everything apart from the part of the proof that $\displaystyle \varphi_{m}$ is surjective.

Given $\displaystyle x\in M$, define:

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

Fernando Revilla

3. Great, makes perfect sense, thanks!

4. Originally Posted by MaximalIdeal
Great, makes perfect sense, thanks!

You are welcome

Fernando Revilla