TMn-Coherent Modules and TMn-Flat Modules

Abstract

In this paper, with respect to a tilting module $T$, the notions of $T_M^n$-coherence and $T_M^n$-flatness are introduced, for every module $M$ and every nonnegative integer $n$. Some characterizations of $T_M^n$-coherent modules are proved. We  show that an $R$-module $F$ is $T_M^n$-flat (injective) if and only if $F$ is $T_{Rm}^n$-flat (injective), for any $m\in M$. Also, some sufficient conditions under which any direct product (direct limit) of $T_M^n$-flat ($T_M^n$-injective) modules is $T_M^n$-flat ($T_M^n$-injective) are given. Among other results, $T_M^n$-coherent rings are studied.