Morphism from Ring with Unity to Module

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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 $m$.

Proof

Existence

Let $r \in R$.

Let $\map \psi r := r m$.

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

$\Box$

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:

$\map \ker {\psi_1 - \psi_2} = R$

and:

$\psi_1 = \psi_2$

$\blacksquare$