\begin{definition}[Definition:Compatible Module Structures]
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 $B$
\end{definition}