ON n-FI-INJECTIVE AND n-FI- FLAT MODULES

Let R be a ring, n a fixed non-negative integer and FIn (F-n) the class of all left (right) R-modules of FP-injective (flat) dimension at most n. A left R-module M (resp., right R-module N) is called n-FI-injective (resp., n-FI-flat) if Ext(R)(1)(F, M) = 0 (resp., Tor(1)(R)(N, F) = 0) for any F is an element of FIn. It is proved that a left R-module M is n-FI-injective if and only if M is a kernel of an FIn-precover f : A -> B of a left R-module B with A injective. For a left coherent ring R, it is shown that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an F-n-preenvelope K -> F of a right R-module K with F flat. Some known results are extended. Finally, we investigate n-FI-injective and n-FI-flat modules over left coherent rings with FP-id(R-R) <= n.