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

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.