The Grothendieck ring of finite-dimensional vector spaces

Dimension is additive on short exact sequences: a basis of a subspace, together with lifts of a basis of the quotient, is a basis of the middle space. Thus $d([V])={\dim }_{k}V$ respects every relation in [F2] and induces a group homomorphism $d:K_0\to \mathbb{Z}$. Split direct-sum sequences and [F1] give $[V]=({\dim }_{k}V)[k]$, including $[0]=0$. Hence $e:\mathbb{Z}\to K_0$, $m\mapsto m[k]$, satisfies $de=\mathrm{id}$ and $ed=\mathrm{id}$, proving the asserted additive bijection on all classes, including negative virtual classes.

Transport integer multiplication along this bijection: set $a\cdot b=e(d(a)d(b))$. This is a well-defined unital ring structure, with unit $e(1)=[k]$. Tensor products of finite bases give ${\dim }_k(V{\otimes }_{k}W)={\dim }_{k}V{\dim }_{k}W$, so $[V]\cdot [W]=e({\dim }_k(V{\otimes }_{k}W))=[V{\otimes }_{k}W]$. Thus the intended tensor multiplication in [F2] descends to the quotient; its bilinear extension is unique because object classes generate the group. The map $d$ is therefore an isomorphism of rings, with zero and unit preserved.