Single-object additive category

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.

2 Answers

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).

• Ah, thanks I had forgotten the empty biproduct (aka zero object). – Bruno Le Floch Aug 7 '16 at 8:02

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.

• You can find an example of a ring having this property at en.wikipedia.org/wiki/Invariant_basis_number – Mariano Suárez-Álvarez Aug 6 '16 at 18:41
• Thank you, I didn't know of such examples. However you made the same mistake I made, of only considering non-empty finite biproducts (well, binary biproduct but this is enough). See the other answer: the empty biproduct forces $\bullet$ to be initial (and terminal), hence there must be a single morphism. – Bruno Le Floch Aug 7 '16 at 8:06