I am interested in finding a link between the preadditive categories when we talk about generalization of the rings with the semigroupoid when we talk about of generalization of near-rings.
Is there a structure or a category that explains the difference that binds and join in a more abstract set, from the categorical point of view, the rings as preadditive categories with near-rings as semigroupoids?
=> A preadditive category is a category enriched over the category Ab of Abelian groups, where composition of morphisms is bilinear. A ring is then a preadditive category with a unique object.
Thus, a near-ring is a semigroupoid (rather than a category) of one unique object enriched over the category $Grp$ of groups, where composition of morphisms is linear. That is, $(f+g)∘h=f∘h+g∘h$ (if right-linear). The generalization just removes the “unique object” restriction.
I don’t know if this has a name in the literature, but I guess we could call such a structure a “colax Grp-semigroupoid”.
PS: I don't understand well but what is also difference between near-ring modules and quasi-modules ?