----- > [!proposition] Proposition. ([[every abelian group is a Z-module, in exactly one way]]) > Given an [[abelian group]] $M$, the [[ring]] $\mathbb{Z}$ [[module|acts on]] $M$ in exactly one way. > That is, there is a unique [[ring homomorphism]] $\sigma: \mathbb{Z} \to \text{End}_{\mathsf{Ab}}(M)$ from $\mathbb{Z}$ into the [[endsets in ab are rings|endomorphism ring]] of $M$. > ^proposition > [!proof]+ Proof. ([[every abelian group is a Z-module, in exactly one way]]) > This is just because [[ring#^properties|the integers are]] [[terminal object|initial]] in the [[category]] $\mathsf{Ring}$. > > Concretely, $\sigma$ takes $n \in \mathbb{Z}$ to the function $a \mapsto \overbrace{\id(a) + \dots + \id(a)}^{n \text{ times}}$, i.e., $a \mapsto na$. In the form of the 'explicit' version of the ring action definition, the $\mathbb{Z}$-[[module]] structure looks like $\begin{align} \mathbb{Z} \times M \to& M \\ (n, a) \mapsto& \sigma(n)(a)=na. \end{align}$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```