- #1

Kreizhn

- 743

- 1

## Homework Statement

Let R be a commutative ring, and M be an R-module. Show that

[tex] \text{Hom}_{\text{R-mod}}(R,M) \cong M [/tex]

as R-modules, where the homomorphisms are R-module homomorphisms.

## The Attempt at a Solution

This should hopefully be quick and easy. The most natural mapping to consider is

[itex] \phi: \text{Hom}_{\text{R-mod}}(R,M) \to M [/itex] sending [itex] f \to f(1_R) [/itex]. It is simple to show that this is a R-mod homomorphism, and that it is injective. Where I am stuck is surjectivity.

The first thing that comes to mind is that I want to use constant maps; however, these are not R-mod homs. Secondly, I've realized that I have not yet used the fact that the ring is commutative. I'm wondering if somehow f(rs) = rf(s) = sf(r) comes into play.