Lemma 2

Let $A$ be a ring, $T$ an $A$-module and $(T_i)$ a right directed family of submodules of $T$ with union $T$. Then $\operatorname{long}_A(T) = \sup_i (\operatorname{long}_A(T_i))$.

$\operatorname{long}_A(T_i) \leq \operatorname{long}_A(T)$ for all $i$. The lemma is obvious if no integer exceeds the $\operatorname{long}_A(T_i)$, both sides then being infinite. Otherwise, let $i_0$ be an index for which $\operatorname{long}_A(T_{i_0})$ takes its greatest value; then $T_{i_0} = T$ since the family $(T_i)$ is directed; whence our assertion in this case.