Let $R$ be an arbitrary ring. Let $D$ be the class of $R$-modules of projective dimension less than or equal to a natural number $n$. If $L$ is the direct union of a continuous chain of submodules ${L_{\alpha},\alpha < \lambda}$ for some ordinal number $\lambda$ (this means that $L=\bigcup_{\alpha}L_{\alpha},\ L_{\alpha}\subseteq L_{\alpha'}$ if $\alpha \leq \alpha' <\lambda$ and $\ L_{\beta}=\bigcup_{\alpha <\beta} L_\alpha$ when $\beta < \lambda $ is a limit ordinal) with $L_{0}\in D$ and $L_{\alpha +1}/L_{\alpha}\in D, \forall \alpha<\lambda,$ can one show that $L \in D$?

PS: We know that when $R$ is a perfect ring, then $D$ is closed under direct limits, then we can prove the above by transfinite induction. But if $R$ is not perfect, how can we show that?