Definition 1

We say that the set $\mathscr{C}$ of classes of A-modules is additive if every zero module is of type $\mathscr{C}$ and the direct sum of two modules of type $\mathscr{C}$ is of type $\mathscr{C}$. We say that $\mathscr{C}$ is hereditary if it is additive and the submodules and quotient modules of a module of type $\mathscr{C}$ are of type $\mathscr{C}$.