Proposition 7

A left A-submodule of $A^*$ of finite dimension over K is contained in $\Theta_{\mathscr{C}}(A)$ if and only if it is of type $\mathscr{C}$.

Let V be a left A-submodule of $A^*$ of finite dimension over K. We saw in VIII, p. 379, Remark 3 that we have $V\subset \Theta_V(A)$. If V is of type $\mathscr{C}$, then we have $\Theta_V(A)\subset \Theta_{\mathscr{C}}$ (A), so V is contained in $\Theta_{\mathscr{C}}(A)$. Conversely, suppose that V is contained in $\Theta_{\mathscr{C}}(A)$. Since $\Theta_{\mathscr{C}}(A)$ is the union of the directed family $(\Theta_E(A))_{E\in\mathscr{C}}$ and V is finite-dimensional over K, there exists a module E of type $\mathscr{C}$ such that V is contained in $\Theta_E(A)$. Now, there exists a natural number $n$ such that $\Theta_E(A)$ is isomorphic to a quotient of $E^n$ (VIII, p. 378, Proposition 2). Since $\mathscr{C}$ is hereditary, V is of type $\mathscr{C}$.