Proposition 6

Let $\mathscr{S}_0$ be the set of classes of simple A-modules of finite dimension over K with nonzero trace. The family of linear forms (Tr$_S$)$_{S\in\mathscr{S}_0}$ is free over K.

Let F be a finite subset of $\mathscr{S}_0$, and let S be an element of F. By assumption, there exists an element $a\in A$ such that Tr$_S(a)\not= 0$. By Corollary 1 of Proposition 4 (VIII, p. 83), there exists an element $b\in A$ such that $b_S=a_S$ and $b_T= 0$ for every $T\in F-\{S\}$. We have Tr$_S(b)\not= 0$ and Tr$_T(b) = 0$ for $T\in F-\{S\}$. The family (Tr$_S$)$_{S\in F}$ is therefore free. Proposition 6 follows.