TITLE: Differences between groups and rings when understood as categories
QUESTION [8 upvotes]: A group can be understood as a category with a single object $*$ whose hom-set $\text{hom}(*,*)$ forms a group with composition as the group multiplication.
A ring can be understood as a category that is preadditive with a single object $*$. Preadditive means that for a category $C$, $\text{hom}_C(A,B)$ forms an abelian group that is distributive with respect to composition of arrows.
The group structure appears in different ways in these two categories. It seems weird to me that the group multiplication is the composition of arrows for groups as categories but for rings it is some binary map of arrows in the same hom-set. Why is this the 'best' way to understand these structures as categories? Is there some way to understand a ring as a category where composition of arrows forms a group (like the categorical sense of a group)?

REPLY [4 votes]: I don’t know if this quite answers your question, but the point you focus on as your dissatisfaction seems to me to be a misunderstanding:

It seems weird to me that the group multiplication is the composition of arrows for groups as categories but for rings it is some binary map of arrows in the same hom-set.

Viewing a ring as a one-object pre-additive (i.e. $\newcommand{\AbGp}{\mathrm{AbGp}}\AbGp$-enriched) category, its multiplication is precisely composition of arrows, exactly like in groups-viewed-as-one-object categories.  The point is that in single-object categories (plain or enriched), composition is exactly “some binary map of arrows in the same hom-set”, $\newcommand{\C}{\mathcal{C}}\C(*,*) \otimes \C(*,*) \to \C(*,*)$.
So I’d say the picture you set out is indeed a very good way to look at them:

monoids (so in particular, groups) are 1-object categories;
rings are 1-object categories enriched in $(\AbGp,\otimes)$ (aka monoids in $(\AbGp,\otimes)$);
the forgetful functor $(\AbGp,\otimes) \to (\mathrm{Set},\times)$ is lax monoidal; so it sends monoids in $(\AbGp,\otimes)$ to monoids in $(\mathrm{Set},\times)$, sending a ring to its underlying multiplicative monoid.