# Definition:Compatible Module Structures/Definition 1

## 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 $a \in A$, $b \in B$, the homotheties $h_a$ and $h_b$ commute.

That is, for all $m \in M$, $a \in A$, $b \in B$:

$a * \paren {b \circledast m} = b \circledast \paren {a * m}$