How do we construct a homomorphism between a simple R-module and a unitary R-module? I don't have a lot of experience with this sort of thing and so I am trying to synthesize a process by which I can understand constructing homomorphisms. In this particular case I want to construct a homomorphism between a simple $R$-module and a unitary $R$-module, where $R$ itself is a ring with identity, where $I_{R} \neq 0_{R}$.
I do know that a non-zero unitary $R$-module $M$ is simple if the only submodules of $M$ are $\left\{0_{R}\right\}$ and $M$. Any suggestions on how to go about this?

Hints:
(1) A simple module $M$ must be cyclic. That is, $M = Rm = \{rm \mid r \in R \}$ for any $m \neq 0 \in M$. Why?
(2) For any module homomorphism $\varphi: M \to N$,
$$
\ker \varphi = \{ m \in M \mid \varphi(m) = 0 \}, 
$$
is a submodule. Why?
(3) A homomorphism $\varphi: M \to N$ is completely determined by $\varphi(m)$, where $m$ is any nonzero element (generator) of $M$. Why?
(4) There are only two possibilities for the image submodule
$$
\operatorname{im} \varphi = \{ \varphi(m) \mid m \in M\}.
$$
What are they? Why?
