Any abelian category$C$ gives rise to an abelian group $K(C)$ called its Grothendieck group (see there for more). If we apply this construction to a monoidal abelian category or generally to a rig category, $K(C)$ is a ring, called the Grothendieck ring.

Let $C$ be an abelian category. The Grothendieck group$K(C)$ is the abelian group generated by objects of $C$, quotiented by the relation that $[A]+[B]=[C]$ if there is a short exact sequence

$0\to A\to C\to B.$

In the presence of a monoidal structure $\otimes$, we can turn $C$ into a ring by defining

$[A]\cdot [B]=[A\otimes B].$

This endows $K(C)$ with the structure of a ring because