Communications of the
Korean Mathematical Society CKMS

This paper introduces and studies a generalization of $(n,d)$-rings introduced and studied by Costa in 1994 to rings with prime nilradical. Among other things, we establish that the $\phi$-von Neumann regular rings are exactly either $\phi$-$(0,0)$ or $\phi$-$(1,0)$ rings and that the $\phi$-Pr\"ufer rings which are strongly $\phi$-rings are the $\phi$-$(1,1)$ rings. We then introduce a new class of rings generalizing the class of $n$-coherent rings to characterize the nonnil-coherent rings introduced and studied by Bacem and Benhissi.

Keywords: Nonnil-coherent ring, $\phi$-Noetherian ring, $\phi$-$n$-presented module, nonnil-FP-injective module, $\phi$-$(n,d)$-injective modules, $\phi$-$(n,d)$-flat modules, $\phi$-$(n,d)$-ring, $\phi$-weak-$(n,d)$-ring, $\phi$-$n$-coherent ring, $\phi$-$n$-von Neumann regular ring

MSC numbers: 13A02, 13A15

Supported by: H. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (2021R1I1A3047469).