nLab
Grothendieck ring

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.

If $C$ is a braided monoidal category, $K(C)$ becomes a commutative ring.

If $C$ is a symmetric monoidal category, $K(C)$ becomes a $\Lambda$-ring — even better.

If $C$ is just braided monoidal, is $K(C)$ just a commutative ring?