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.

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