Definition:Compatible Module Structures/Definition 1
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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 $m \in M$, $a \in A$, $b \in B$:
- $a * \paren {b \circledast m} = b \circledast \paren {a * m}$