Some Remarks on $ϕ$-Dedekind rings and $ϕ$-Prufer rings

In this paper, the notions of nonnil-injective modules and nonnil-FP-injective modules are introduced and studied. Especially, we show that a $ϕ$-ring $R$ is an integral domain if and only if any nonnil-injective (resp., nonnil-FP-injective) module $R$-module is injective (resp., FP-injective). Some new characterizations of $ϕ$-von Neumann regular rings, nonnil-Notherian rings and nonnil-coherent rings are given. We finally characterize $ϕ$-Dedekind rings and $ϕ$-\Prufer\ rings in terms of $ϕ$-flat modules, nonnil-injective modules and nonnil-FP-injective modules.

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