Relative copure injective and copure flat modules

Let R be a ring, n a fixed nonnegative integer and image (image) the class of all left (right) R-modules of injective (flat) dimension at most n. A left R-module M (resp., right R-module F) is called n-copure injective (resp., n-copure flat) if image (resp., image) for any image. It is shown that a left R-module M over any ring R is n-copure injective if and only if M is a kernel of an image-precover f:A→B of a left R-module B with A injective. For a left coherent ring R, it is proven that every right R-module has an image-preenvelope, and a finitely presented right R-module M is n-copure flat if and only if M is a cokernel of an image-preenvelope K→F of a right R-module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.