On n-FI-Injective and n-FI-Flat Modules

Abstract

Let R be a ring, n a fixed non-negative integer and ℱℐ n (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 (resp., ) for any F ∈ ℱℐ n . It is proved that a left R-module M is n-FI-injective if and only if M is a kernel of an ℱℐ n -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.