Definition:Linear Ring Action

Definition
Let $R$ be a ring.

Let $M$ be an abelian group.

Left ring action
A (left) linear ring action of $R$ on $M$ is a mapping from the cartesian product $\circ : R \times M \to M$ such that:

Right ring action
A right linear ring action of $R$ on $M$ is a mapping from the cartesian product $\circ : M \times R \to M$ such that:

Also known as
A left ring action is also known as a ring action.

Also see

 * Definition:Module over Ring
 * Correspondence between Linear Ring Actions and Ring Representations