Some remarks on nonnil-coherent rings and $\phi$-IF rings

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.