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.