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

Abstract

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

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