Corollary

Let E be an A-module of finite dimension over K. Then E is of type $\mathscr{C}$ if and only if $\Theta_E(A)$ is contained in $\Theta_{\mathscr{C}}(A)$.

The condition is obviously necessary.

Conversely, suppose that $\Theta_{\mathscr{C}}(A)$ contains $\Theta_E(A)$. The A-module $\Theta_E(A)$ is finite-dimensional over K (VIII, p. 378, Proposition 2), so it is of type $\mathscr{C}$ by Proposition 7. Now, there exists an integer $n\geqslant 0$ such that E is isomorphic to an A-submodule of $\Theta_E(A)^n$ (VIII, p. 378, Proposition 2). Since $\mathscr{C}$ is hereditary, E is of type $\mathscr{C}$.