Definition:Compatible Module Structures/Definition 3

Definition
Let $A$ and $B$ be rings.

Let $\struct {M, +}$ be an abelian group.

Let $* : A \times M \to M$ and $\circledast: B \times M \to M$ be left or right linear ring actions so that:
 * $(1): \quad \struct {M, +, *}$ is a left or right module over $A$
 * $(2): \quad \struct {M, +, \circledast}$ is a left or right module over $B$

The module structures are compatible for all $b \in A$, the homothety $h_b : M \to M$ is an endomorphism of the $A$-module $M$.

That is, the image of the ring representation $B \to \map {\operatorname {End} } M$ is contained in the endomorphism ring $\map {\operatorname {End}_A} M$.

Also see

 * Equivalence of Definitions of Compatible Module Structures