Definition:Compatible Module Structures

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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 $B$


Definition 1

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 $m \in M$, $a \in A$, $b \in B$:

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


Definition 2

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

That is, if and only if the image of the ring representation $A \to \map {\operatorname {End} } M$ is contained in the endomorphism ring $\map {\operatorname {End}_B } M$.


Definition 3

The module structures are compatible if and only if for all $b \in A$, the homothety $h_b : M \to M$ is an endomorphism of the $A$-module $M$.

That is, if and only if the image of the ring representation $B \to \map {\operatorname {End} } M$ is contained in the endomorphism ring $\map {\operatorname {End}_A} M$.


Also see