Definition:Z-Module Associated with Abelian Group
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Definition
Let $\struct {G, *}$ be an abelian group with identity $e$.
Let $\struct {\Z, +, \times}$ be the ring of integers.
Definition 1
The $\Z$-module associated with $G$ is the $\Z$-module $\struct {G, *, \circ}$ with ring action:
- $\circ: \Z \times G \to G$:
- $\tuple {n, x} \mapsto *^n x$
where $*^n x$ is the $n$th power of $x$.
Definition 2
The $\Z$-module associated with $G$ is the $\Z$-module on $G$ with ring representation $\Z \to \map {\operatorname {End} } G$ equal to the initial homomorphism.
Also see
- Equivalence of Definitions of Z-Module Associated with Abelian Group
- Z-Module Associated with Abelian Group is Unitary Z-Module
- Correspondence between Abelian Groups and Z-Modules
- Results about the $\Z$-module associated with an abelian group can be found here.