Relative FI-injective and FI-flat modules

Abstract

Let R be a ring, n a fixed nonnegative integer and FPn (Fn) the class of all left (right) R-modules of FP-injective (flat) dimensions at most n. A left R-module M (resp., right R-module F) is called n-FI-injective (resp., n-FI-flat) if Ext1(N,M) = 0 (resp., Tor1(F,N) = 0) for any N ∈ FPn. It is shown that a left R-module M over any ring R is n-FI-injective if and only if M is a kernel of an FPn-precover f: A → B with A injective. For a left coherent ring R, it is proven that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an Fn-preenvelope K → F of a right R-module K with F projective if and only if M ∈⊥Fn. These classes of modules are used to construct cotorsion theories and to characterize the global dimension of a ring.