Abelian groups as $ℤ$ -modules

The notions of abelian group and $ℤ$ -module agree, as well as the corresponding notions of homomorphism. In fact, $𝖠 𝖻$ and $ℤ 𝖬 𝗈 𝖽$ are isomorphic categories. #m/thm/module

Proof

First note that every $ℤ$ -module is an abelian group under addition by definition, and every $ℤ$ -linear map is a group homomorphism. Thus there exists a “forgetful functor” (which turns out not to be forgetting anything)

𝐹 : ℤ 𝖬 𝗈 𝖽 \to 𝖠 𝖻

Now every abelian group $𝐴$ admits a $ℤ$ -action, where for $𝑎 \in 𝐴$ and $𝑛 \in ℤ$ we say

𝑛 \cdot 𝑎 = ⎧ { {⎨ { {⎩ \sum 𝑛 𝑎 𝑛 > 0 \sum 𝑛 - 𝑎 𝑛 < 0 0 𝑛 = 0

which is easily verified to satisfy all properties of a $ℤ$ -module. Thus we have a functor

𝐺 : 𝖠 𝖻 \to ℤ 𝖬 𝗈 𝖽

where $𝐹 𝐺 = 1 𝖠 𝖻$ and $𝐺 𝐹 = 1 ℤ 𝖬 𝗈 𝖽$ , hence the categories are isomorphic.

#state/tidy | #lang/en | #SemBr

Abelian groups as ℤ-modules

Abelian groups as $ℤ$ -modules