R-linear categories Wikipedia "Preadditive category" and nLab "algebroid" seem to define $R$-linear category as one whose hom-sets are $R$-modules, where those modules are over a COMMUTATIVE ring. Does anybody knows why they require the ring to be commutative? When defining preadditive categories, categories whose hom-sets are abelian groups, no such requirement is made. What is the difference? I anticipate the objection that abelian groups are modules over $\mathbb Z$ which is commutative too. But Wikipedia in the same article, includes the category of (left) modules over a ring $R$ as an example of a preadditive category, and that ring is not required to be commutative.

The source of the reason is in module theory.  For $R$-modules $M$ and $N$, the set $\mathrm{Hom}_R(M,N)$ is naturally an abelian group.  You'd prefer it to be an $R$-module since that's the category from which $M$ and $N$ came.  This is true if $R$ is commutative.  In particular, if $M$ is an $(R,S)$-bimodule then 
$\mathrm{Hom}_R(M,N)$ has a natural $S$-module structure, and there is a similar story in the $N$-variable.  But, if $R$ is commutative, everything works out to be the same in both variables and (a) $\mathrm{Hom}_R(M,N)$ is an $R$-module and (b) this structure is compatible in both slots.  
