Proposition 7

Let $C$ be a commutative ring and for every matrix $P \in \mathbf{M}_n(C)$ let $f_P$ be the linear form $X \mapsto \operatorname{Tr}(PX)$ on $\mathbf{M}_n(C)$; the mapping $P \mapsto f_P$ is a $C$-linear bijection of $\mathbf{M}_n(C)$ onto its dual.