Remark 2

Let E be an A-module, and let $E'$ be a submodule of E. By the corollary of Theorem 5 of II, §7, No. 5, p. 299, we have $\Theta_{E'}(A)\subset \Theta_E(A)$ and $\Theta_{E/E'}(A)\subset \Theta_E(A)$.

Let $\gamma_E$ be the unique K-linear mapping from $E\otimes_KE^*$ to $A^*$ that sends $x\otimes x^*$ to $c_E(x, x^*)$. It is $(A$, A)-bilinear, and its image is $\Theta_E(A)$. We deduce from it A-linear mappings $c'_E: E\rightarrow$ Hom$_A(E^*,A^*)$ and $c''_E: E^*\rightarrow$ Hom$_A(E,A^*)$ such that

Denote by $\theta_E$ the K-linear mapping $\theta_E: E\otimes E^*\rightarrow$ End$_K(E)$ characterized by the relation

for $x, y\in E$ and $x^*\in E^*$. Its image is the set End$^f_K(E)$ of endomorphisms of E of finite rank (VIII, p. 463). By the definition of the trace (loc. cit.), we have

Tr($\theta_E(x\otimes x^*)$) $=\langle x^*, x\rangle$

for $x\in E$ and $x^*\in E^*$; we therefore have

$\langle \gamma_E(x\otimes x^*), a\rangle =\langle x^*, ax\rangle =$ Tr($\theta_E(ax\otimes x^*)$) $=$ Tr($\theta_E(x\otimes x^*)a$).

This gives the relation

$\langle \gamma_E(h), a\rangle =$ Tr($\theta_E(h)\circ a_E$)

for $a\in A$ and $h\in E\otimes_KE^*$. This proves that $\Theta_E(A)$ is the set of linear forms $a\mapsto$ Tr($u\circ a_E$) on A, where $u$ runs through End$^f_K(E)$.