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