Rings which have \((m, n)\)-flat injective modules

Abstract

A  ring  \(R\)  is  said  to  be  a left \(IF-(m,n)\) ring if every  injective  left \(R\)-module is \((m,n)\)-flat. In this paper, several characterizations of left \(IF-(m,n)\) rings are investigated, some conditions under which \(R\) is left \(IF-(m,n)\) are given. Furthermore, conditions under which a left \(IF-1\) ring (i.e., \(IF-(1,1)\) ring) is a field, a regular ring and a semisimple ring are studied respectively.