Question

# Let R be a commutative ring. Prove that Hom_R(R, M) and M are isomorphic R-modules

Commutative Algebra
Let R be a commutative ring. Prove that $$\displaystyle{H}{o}{m}_{{R}}{\left({R},{M}\right)}$$ and M are isomorphic R-modules

2021-01-20
Define $$\displaystyle{F}:{H}{o}{m}{\left({R},{M}\right)}\to{M}$$ by $$\displaystyle{F}{\left(\phi\right)}=\phi{\left({1}\right)}\in{M}$$
First, we proved that F is an R-module homomorphism.
Let $$\displaystyle\phi,\rho\in{H}{o}{m}{\left({R},{M}\right)}{\quad\text{and}\quad}{r}\in{R}$$. Then,
$$\displaystyle{F}{\left({r}\phi-\rho\right)}={\left({r}\phi-\rho\right)}{\left({1}\right)}={r}\phi{\left({1}\right)}-\rho{\left({1}\right)}={r}{F}{\left({r}\right)}-{F}{\left({r}\right)}$$
Hence, F is an R-module homomorphism.
Now, to prove that it is an isomorphic R-module, we must prove that F is bijective.
Suppose $$\displaystyle{F}{\left(\phi\right)}={0}$$ for some $$\displaystyle\phi\in{H}{o}{m}{\left({R},{M}\right)}$$. Then, $$\displaystyle\phi{\left({1}\right)}={0}.$$
For any $$\displaystyle{r}\in{R},\phi{\left({r}\right)}={r}\phi{\left({1}\right)}={r}\cdot{0}={0}$$. Hence, $$\displaystyle\phi={0}$$. Thus, F is injective.
Now, suppose $$\displaystyle{m}\in{M}$$.
Define a map $$\displaystyle\phi:{R}\to{M}$$ by $$\displaystyle\phi{\left({r}\right)}={r}{m}$$ for any $$\displaystyle{r}\in{R}.$$
Let $$\displaystyle{r},{s},{t}\in{R}.$$
$$\displaystyle\Rightarrow\phi{\left({r}{s}-{t}\right)}={\left({r}{s}-{t}\right)}{m}={r}{\left({s}{m}\right)}-{t}{m}={r}\phi{\left({s}\right)}-\phi{\left({t}\right)}$$
Hence, $$\displaystyle\phi\in{H}{o}{m}{\left({R},{M}\right)}.$$
Futher, F $$\displaystyle{\left(\phi\right)}=\phi{\left({1}\right)}={m}.$$
Thus, F is surjective.
F is homomorphism and bijective. Therefore, $$\displaystyle{H}{o}{m}{\left({R},{M}\right)}{\quad\text{and}\quad}{M}$$ are isomorphic R-module.