FI_n-injecive and FI_n-flat modules over commutative rings

Let R be a commutative ring, n \gt 1 an integral number. FI_n-injecive and FI_n-flat modules are introduced. We prove that an R -module M is FI_n-injecive if and only if M is a kernel of an FP_n-injecive precover f\colon A\rightarrow B with A is injective. It indicates that an R -module M is a reduced FI_n-injective module if and only if M is a kernel of an FP_n-injecive cover f\colon A\rightarrow B with A is injective. We obtain that an R -module M is FI_n-injecive if and only if M is a direct sum of an injective R -module and a reduced FI_n-injective module. Finally, we give the sufficient and necessary conditions when every R -module is FI_n-flat, if R is n -hereditary.