Module theory MOC

Abelian groups as ℤ-modules

The notions of abelian group and ℤ-module agree, as well as the corresponding notions of homomorphism. In fact, Category of abelian groups and [[Category of modules over a commutative ring|ℤ𝖬𝗈𝖽]] are isomorphic categories. 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:ℤ𝖬𝗈𝖽→𝖠𝖻

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

𝑛⋅𝑎=⎧{ {⎨{ {⎩∑𝑛𝑎𝑛>0∑𝑛−𝑎𝑛<00𝑛=0

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

𝐺:𝖠𝖻→ℤ𝖬𝗈𝖽

where 𝐹𝐺 =1𝖠𝖻 and 𝐺𝐹 =1ℤ𝖬𝗈𝖽, hence the categories are isomorphic.

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