Definition:Compatible Module Structures/Definition 2

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 if and only if for all $a \in A$, the homothety $h_a : M \to M$ is an endomorphism of the $B$-module $M$.

That is, if and only if the image of the ring representation $A \to \map {\operatorname {End} } M$ is contained in the endomorphism ring $\map {\operatorname {End}_B } M$.