Remark

Under only assumptions (i) and (ii) of Proposition 9, the Grothendieck group $K'(\mathscr{C})$ for direct sums (VIII, p. 188) has a unique ring structure whose multiplication satisfies $[E]'_{\mathscr{C}}[F]'_{\mathscr{C}}= [E\otimes F]'_{\mathscr{C}}$. Its unit element is $[K_{\gamma}]'_{\mathscr{C}}$. The ring $K'(\mathscr{C})$ is commutative if the bigebra A is cocommutative. The proof is analogous to that of Proposition 9 because the group $K'(\mathscr{C})$ can be identified with $\mathbf{Z}^{(\mathscr{C})}/R'$, where $R'$ is the subgroup of $\mathbf{Z}^{(\mathscr{C})}$ generated by the elements of the form cl(E$'\oplus E''$)$-$ cl(E$'$)$-$ cl(E$''$) (loc. cit.).

Under assumptions (i), (ii), and (iii) of Proposition 9, the ring $K(\mathscr{C})$ is called the Grothendieck ring of $\mathscr{C}$. These conditions are, in particular, satisfied when K is a field and $\mathscr{C}$ is the set of classes of A-modules that are finite-dimensional over K. Consequently, we have the following.