A module over PID

Question

Let R be a PID and M be a module over R ⋀ cyclic. Then prove that ∀N⊂M; a submodule of M, N is cyclic.

[My idea]

∃m∈M s.t. M=Rm (∵M is cyclic)

p: R(= a module over R) → M, p(a) = am; module-hom ⋀ surjective.

Now, let ∀N⊂M; a submodule of M be fixed. I tried to use "theorem on homomorphisms of modules", but I don't have no idea about next:

--how use this theorem? (What is Ker(p), Im(p)? What should I divide R by?) --When should I use assumption: R is PID ?

Show me the careful answer, please.

Answer 1

Let $N$ be a submodule of $M=Rm$ and consider

$I:=\{r\in R : rm\in N\}$.

Then $I$ is an ideal of $R$ (check this). Hence $I=Rx$ for some $x\in R$. Consequently $N=Rxm$.