Definition:Compatible Module Structures/Definition 2
Jump to navigation
Jump to search
This article needs to be linked to other articles. In particular: "module structures" You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
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 $a \in A$, 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 $\map {\operatorname {End}_B } M$.