Category:Definitions/Compatible Module Structures
This category contains definitions related to Compatible Module Structures.
Related results can be found in Category:Compatible Module Structures.
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 $\map {\operatorname {End}_A} M$.
Pages in category "Definitions/Compatible Module Structures"
The following 4 pages are in this category, out of 4 total.