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

Abstract

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