I was told that commutative ring category is not additive category all the time and it is thus not abelian category.
A category is additive if $Hom(A,B)$ is abelian group for all objects $A,B$ and morphisms satisfy distributive laws, it has zero objects and it has finite coproduct and products.
It seems commutative ring satisfies most of the requirement except $Hom(A,B)$ being abelian group.
$Hom(A,B)$ is not abelian group by taking any $f\in Hom(A,B)$. If $f$ has inverse $g$, I can check $g(a_1a_2)=f(a_1a_2)$ which never cancels out with $f(a_1a_2)$.
It definitely satisfies distributive laws of morphisms.(Wrong. Distributive law fails due to not fixing $1$.)
Zero object is 0 ring as identity element is $0$ in 0 ring.
Coproduct is tensor and product is direct sum.
Does commutative ring category fail only $Hom(A,B)$ being abelian group requirement?