## Every ring action induces a group action

Let $R$ be a ring with 1 and let $M$ be a left unital $R$-module. Prove that the restriction of the module operator $\cdot$ to $R^\times \times M$ is a group action operator on the group of units in $R$.

Recall that $R^\times$ is indeed a group; certainly the restriction of $\cdot$ to $R^\times \times M$ is a mapping $R^\times \times M \rightarrow M$. Now for all $m \in M$, we have $1 \cdot m = m$, and if $u,v \in R^\times$, then $(uv) \cdot m = u \cdot (v \cdot m)$. So $R^\times$ acts on $M$ by the restriction of our module operator $\cdot$.