Definition:Compatible Module Structures

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Definition

Let $A$ and $B$ be rings.

Let $\left({M, +}\right)$ 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:

$(1): \quad \left({M, +, *}\right)$ is a left or right module over $A$
$(1): \quad \left({M, +, \circledast}\right)$ 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 $m \in M$, $a \in A$, $b \in B$:

$a * \left({b \circledast m}\right) = b \circledast \left({a * m}\right)$


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 $A \to \operatorname{End} \left({M}\right)$ is contained in the endomorphism ring $\operatorname{End}_B \left({M}\right)$.


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 $B \to \operatorname{End} \left({M}\right)$ is contained in the endomorphism ring $\operatorname{End}_A \left({M}\right)$.


Also see