My motivation comes from How to see that a one-object pre-additive category is a ring? and this master thesis. Basically, I want to see concrete examples of preadditive categories and what structure they induce.
The basic example
A preadditive category $\mathcal{A}$ with $obj(\mathcal{A}) = \{X\}$ induces a ring structure in the sets of morphisms $(\mathcal{A}(X,X),+,\circ)$.
More advanced example
For a finite number of objects in the category this article Category Theoretic Interpretation of Rings seems to construct a category equivalence with certain kinds of rings.
The infinite case?
However, there are still cases to discuss. What happens if I want to interpret a preadditive category with an infinite number of objects? Someone told me I should obtain a ring without 1. Can you confirm this point?