# Definition:Compatible Module Structures

## 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 $\map {\operatorname {End}_A} M$.