Single-object additive category

Question

A pre-additive category with a single object $\bullet$ is simply a ring $R = \mathrm{Hom}(\bullet,\bullet)$: pre-additivity makes this Hom-space an abelian group and with bilinear composition, i.e. a distributive product.

Out of idle curiosity I am wondering what happens to the ring if we ask for the category to be additive.

Answer 1

Since an additive category has a zero object it follows that an additive category with one object is the trivial ring (i.e. the ring with only one element).

Answer 2

If the category is additive, then $\bullet\oplus\bullet=\bullet$. It follows that $M_2(End(\bullet))\cong End(\bullet)$ as ring. More generally, we have $M_n(End(\bullet))\cong End(\bullet)$ for all $n$. This tells us that thering $End(\bullet)$ is a bit weird... There are examples, though.