ra.rings and algebras - Behavior of the projective dimension of modules in a continuous chain of extensions

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?

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