On n-Coherent Rings and (n,d)-Injective Modules

Abstract

Let R be a ring, n, d be fixed non-negative integers, [Formula: see text] the class of (n,d)-injective left R-modules, and [Formula: see text] the class of (n,d)-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then [Formula: see text] if and only if [Formula: see text], if and only if Ext m+k(M,N) = 0 for all left R-modules M, N and all k ≥ -1, if and only if Ext m-1(M,N) = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then − ⊗ − is right balanced on [Formula: see text] by [Formula: see text], and investigate the global right [Formula: see text]-dimension of [Formula: see text] and the global right [Formula: see text]-dimension of [Formula: see text] by right derived functors of − ⊗ −. Some known results are obtained as corollaries.