\begin{definition}[Definition:Compatible Module Structures/Definition 1]
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''' {{iff}} for all $a \in A$, $b \in B$, the homotheties $h_a$ and $h_b$ commute.
That is, for all $m \in M$, $a \in A$, $b \in B$:
:$a * \paren {b \circledast m} = b \circledast \paren {a * m}$
\end{definition}