\(T_M^n\)-Coherent Modules and \(T_M^n\)-Flat Modules - Shaveisi, Farzad; Amini, Mostafa (Journal of Informatics and Mathematical Sciences, 2017)
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 \(mM\). 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.