Definition:Compatible Module Structures

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

Let $(M, +)$ 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:
 * $(M, +, *)$ is a left or right module over $A$
 * $(M, +, \circledast)$ is a left or right module over $B$

Definition 1
The module structures are compatible for all $a \in A$, $b \in B$, the homotheties $h_a$ and $h_b$ commute. That is, for all $m \in M$, $a \in A$, $b \in B$:
 * $a * (b \circledast m) = b \circledast (a * m)$

Definition 2
The module structures are compatible for all $a \in A$, the homothety $h_a : M \to M$ is an endomorphism of the $B$-module $M$.

That is, the image of the ring representation $A \to \operatorname{End}(M)$ is contained in the endomorphism ring $\operatorname{End}_B(M)$.

Definition 3
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 $B \to \operatorname{End}(M)$ is contained in the endomorphism ring $\operatorname{End}_A(M)$.

Also see

 * Definition:Multimodule