Relative copure injective and copure flat modules

Let RR be a ring, nn a fixed nonnegative integer and InIn (FnFn) the class of all left (right) RR-modules of injective (flat) dimension at most nn. A left RR-module MM (resp., right RR-module FF) is called nn-copure injective (resp., nn-copure flat) if Ext1(N,M)=0 (resp., Tor1(F,N)=0) for any N∈InN∈In. It is shown that a left RR-module MM over any ring RR is nn-copure injective if and only if MM is a kernel of an InIn-precover f:A→Bf:A→B of a left RR-module BB with AA injective. For a left coherent ring RR, it is proven that every right RR-module has an FnFn-preenvelope, and a finitely presented right RR-module MM is nn-copure flat if and only if MM is a cokernel of an FnFn-preenvelope K→FK→F of a right RR-module KK with FF flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.