Definition:Ring Action Defined by Ring Representation

Definition
Let $R$ be a ring.

Let $M$ be an abelian group.

Let $\rho : R \to \operatorname{End}(M)$ be a ring representation.

The associated (left) ring action is the linear ring action:
 * $R \times M \to M$:
 * $(r, m) \mapsto \rho (r)(m)$

Also see

 * Definition:Right Ring Action Defined by Ring Antirepresentation
 * Definition:Ring Representation Defined by Ring Action
 * Correspondence between Linear Ring Actions and Ring Representations