FI-injective and FI-flat modules

Abstract

Let R be a ring. A left R-module M (respectively right R-module N) is called FI-injective (respectively FI-flat) if Ext 1 ( G , M ) = 0 (respectively Tor 1 ( N , G ) = 0 ) for any FP-injective left R-module G. Suppose R is a left coherent ring. It is shown that a left R-module M is FI-injective if and only if M is a direct sum of an injective left R-module and a reduced FI-injective left R-module; a finitely presented right R-module M is FI-flat if and only if M is a cokernel of a flat preenvelope of a right R-module. These modules together with the left derived functors of Hom are used to study the FP-injective dimensions of modules and rings.