Some Remarks on $\varphi$-Dedekind rings and $\varphi$-Prufer rings

Abstract

In this paper, the notions of nonnil-injective modules and nonnil-FP-injective modules are introduced and studied. Especially, we show that a $\varphi$-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 $\varphi$-von Neumann regular rings, nonnil-Notherian rings and nonnil-coherent rings are given. We finally characterize $\varphi$-Dedekind rings and $\varphi$-\Prufer\ rings in terms of $\varphi$-flat modules, nonnil-injective modules and nonnil-FP-injective modules.

Cited by in corpus (1)

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