Envelopes and covers by modules of finite pure-injective and pure-projective dimensions

Let R be a ring and n a fixed non-negative integer. $${\mathscr {C}}$$ C and $${\mathscr {D}}$$ D denote the classes of all left R-modules with pure-injective dimension and pure-projective dimension at most n, respectively. It is proved that for any cardinal number $$\delta $$ δ , there is a cardinal number $$\kappa $$ κ such that for any left R-module $$L \in {\mathscr {C}}$$ L ∈ C (resp. $$L \in {\mathscr {D}}$$ L ∈ D ), and any left R-module $$M\le L$$ M ≤ L with $$|M|\le \delta $$ | M | ≤ δ , there is a pure submodule $$L^{\prime }$$ L ′ of L with $$M\le L^{\prime }$$ M ≤ L ′ , $$|L^{\prime }|\le \kappa $$ | L ′ | ≤ κ and $$L^{\prime }, L/L^{\prime }\in {\mathscr {C}}$$ L ′ , L / L ′ ∈ C (resp. $$L^{\prime }, L/L^{\prime }\in {\mathscr {D}}$$ L ′ , L / L ′ ∈ D ). As applications, it is shown that every left R-module M admits a $${\mathscr {C}}$$ C -preenvelope, and there exists an exact sequence $$0\rightarrow M\rightarrow D\rightarrow D/M\rightarrow 0$$ 0 → M → D → D / M → 0 with $$D\in {\mathscr {D}}^{\bot }$$ D ∈ D ⊥ and $$D/M \in {\mathscr {D}}$$ D / M ∈ D .