Let $R$ be a commutative ring. If the nilpotent radical $\text{N}\text{i}\text{l}(R)$ 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 [Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305], and then characterize nonnil-coherent rings in terms of $\varphi$-flat modules, nonnil-injective 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.

让$R$是一个交换环。如果幂零自由基$\text{ñ}\text{一世}\text{l}(R)$的$R$是一个分裂的素理想，那么$R$被称为$\phi$-戒指。在本文中，我们首先区分非零相干环的类别和$\phi$-Bacem 和 Ali 引入的相干环 [非相干环，Beitr。代数几何。 57 (2) (2016) 297–305]，然后用以下方式表征非零相干环$\phi$-flat 模块、nonnil-injective 模块和 nonnil-FP-injective 模块。一个$\phi$-戒指$R$被称为$\phi$-IF 环，如果任何非零内射模块是$\phi$-平坦的。我们获得了一些模块理论表征$\phi$-IF 响铃。举两个例子来区分$\phi$-IF 环和 IF$\phi$-戒指。