  Let $R$ be a commutative ring. If the nilpotent radical $Nil(R)$ of $R$ is a
divided prime ideal, then $R$ is called a $\phi$-ring. In this paper, we first
distinguish the classes of nonnil-coherent rings and $\phi$-coherent rings
introduced by Bacem and Ali [10], and then characterize nonnil-coherent rings
in terms of $\phi$-flat modules and nonnil-FP-injective modules. A $\phi$-ring
$R$ is called a $\phi$-IF ring if any nonnil-injective module is $\phi$-flat.
We obtain some module-theoretic characterizations of $\phi$-IF rings. Two
examples are given to distinguish $\phi$-IF rings and IF $\phi$-rings.
