\begin{definition}[Definition:Ring Action Defined by Ring Representation]
Let $R$ be a ring.
Let $M$ be an abelian group.
Let $\rho : R \to \map {\operatorname {End} } M$ be a ring representation.
The '''associated (left) ring action''' is the linear ring action:
:$R \times M \to M$:
:$\tuple {r, m} \mapsto \map {\map \rho r} m$
\end{definition}