Definition:Z-Module Associated with Abelian Group/Definition 1
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Definition
Let $\struct {G, *}$ be an abelian group with identity $e$.
Let $\struct {\Z, +, \times}$ be the ring of integers.
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 $x$.
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.7$