\(T_M^n\)-Coherent Modules and \(T_M^n\)-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.