Module theory MOC

Abelian groups as -modules

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

Proof

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

Now every abelian group admits a -action, where for and we say

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

where and , hence the categories are isomorphic.

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