Rescaling is Linear Transformation

Theorem
Let $\left({R, +, \cdot}\right)$ be a commutative ring.

Let $\left({V, +, \circ}\right)_R$ be an $R$-module.

Then for any $r \in R$, the rescaling:


 * $m_r: V \to V, v \mapsto r \circ v$

is a linear transformation.

Proof
Let $v \in V$ and $s \in R$.

Then:

Next, for $v,w \in V$:

It follows that $m_r$ is a linear transformation.