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 [Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297-305], and then characterize nonnil-coherent rings in terms of phi-flat modules, nonnil-injective modules and nonnil-PP-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.
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