# Why the Grothendieck group/ring?

[alg] [cat.theory]
Given an additive category $\mathcal{C}$ we can form the Grothendieck group $\mathcal{K}(\mathcal{C})$. If further $\mathcal{C}$ is a tensor category then $\mathcal{K}(\mathcal{C})$ is also a ring, the Grothendieck ring.

If $\mathcal{C}$ is semisimple then any equality in $\mathcal{K}(\mathcal{C})$ corresponds to an isomorphism in $\mathcal{C}$.

[alg]
Subgroups of finitely-generated abelian groups are finitely-generated.

Not necessarily true for finitely-generated non-abelian groups.

[alg]
If $R$ is a commutative ring then the matrix ring $M_n(R)$ is an associative algebra.

Fails if $R$ is non-commutative: for then we need not have $r(AB) = A(rB)$.