Some results on $(n,d)$-injective modules, $(n,d)$-flat modules and $n$-coherent rings

Abstract

Let $n,d$ be two non-negative integers. A left $R$-module $M$ is called $(n,d)$-injective, if ${\rm Ext}^{d+1}(N, M)=0$ for every $n$-presented left $R$-module $N$. A right $R$-module $V$ is called $(n,d)$-flat, if ${\rm Tor}_{d+1}(V, N)=0$ for every $n$-presented left $R$-module $N$. A left $R$-module $M$ is called weakly $n$-$FP$-injective, if ${\rm Ext}^n(N, M)=0$ for every $(n+1)$-presented left $R$-module $N$. A right $R$-module $V$ is called weakly $n$-flat, if ${\rm Tor}_n(V, N)=0$ for every $(n+1)$-presented left $R$-module $N$. In this paper, we give some characterizations and properties of $(n,d)$-injective modules and $(n,d)$-flat modules in the cases of $n\geq d+1$ or $n> d+1$. Using the concepts of weakly $n$-$FP$-injectivity and weakly $n$-flatness of modules, we give some new characterizations of left $n$-coherent rings.