Morphism from Ring with Unity to Module

Theorem
Let $R$ be a ring with unity.

Let $M$ be an $R$-module.

Then for every $m\in M$ there exists a unique $R$-module morphism


 * $\psi:R\to M$

that sends $1$ to $m$.

Existence
Let $\psi(r) := rm$.

This map is $R$-linear by definition of a module.

Uniqueness
Let $\psi_1$ and $\psi_2$ be two such morphisms.

Then $\psi_1-\psi_2$ is an $R$-module morphism whose kernel contains $1$.

Thus $\ker(\psi_1-\psi_2)=R$, and $\psi_1=\psi_2$.