On nil-Flat Modules, Nil-Injective and nil-Flat Dimensions

This paper give the definition of N-coherent rings, and study the rings by nil-injective modules and nil-flat modules. For an N-coherent ring R , we show that every nil-flat right R- module is flat ⇔ every nil-injective left R- module is FP injective ⇔ every nil-injective pure injective left R-module is injective. Nil-injective dimension and nil-flat dimension are also studied. We show I.nil-id(R) ≤ n ⇔ Extⁱ ( R/Ra, M) = 0 for every i ≥ n and every a ∈ N(R), and every left R-module M ⇔ Extⁿ⁺¹ ( R/Ra, M) = 0 for every a ∈ N(R) and every R-module M ⇔ pd (R/Ra) ≤ n for any a ∈ N(R).