Suppose $A$ is an artin algebra and take the category $\operatorname{mod}A$ of finitely generated $A$-modules. Consider the following construction. Let $M$ be an $A$-module. Since $A$ is artinian, $M$ admits a composition series
$$ 0 = M_1 \subset M_2 \subset \dots \subset M_n = M $$
since $M$ is of finite length. It is well known that each $M_i$ in the composition series is maximal since $M_i/M_{i-1}$ is simple.
Now, consider the full subcategory $\mathcal{C}$ of $\operatorname{mod}A$ generated by all modules that appear in some composition series of an $A$-module. If we do this construction to every $A$-module in $\operatorname{mod}A$, what can we say about this subcategory, does it have a name?
Let $M$ be an $A$-module and consider the inclusion $M \subseteq M\oplus k$ for $k$ the trivial module. Then the inclusion may be refined into a composition series for $M\oplus k$, thus $M\in \mathcal{C}$ and so $\mathcal{C} =$ mod-$A$.