Exercise 1, § 11

If V is a finitely generated projective A-module, so is End(V), canonically identified with $V \otimes_A V^*$ (II, § 4, no. 2, Corollary to Proposition 2). If every element $u \in \mathrm{End}(V)$ is associated with the A-linear form $v \mapsto \mathrm{Tr}(uv)$ on $\mathrm{End}(V)$, an A-linear bijection is defined of $\mathrm{End}(V)$ onto its dual $(\mathrm{End}(V))^*$. Identifying $\mathrm{End}(V)$ and its dual A-module under this mapping, the A-algebra structure on $\mathrm{End}(V)$ defines by duality (no. 1, Example 3) an A-cogebra structure on $\mathrm{End}(V)$, which is coassociative and has as counit the form $\mathrm{Tr}: V \to A$. In particular, for $V = A^n$, a cogebra structure is defined on $\mathbf{M}_n(A)$ for which the coproduct is given by $c(E_{ij}) = \sum_k E_{kj} \otimes E_{ik}$. This cogebra structure and the algebra structure on $\mathbf{M}_n(A)$ do not define a bigebra structure.