Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative ring with nonzero identity. An $R$-module $M$ is said to be $\phi$-P-flat if, for any $s \in R \setminus \operatorname{Nil}(R)$ and any $x \in M$ such that $sx = 0$, we have $x \in (0:s)M$. An $R$-module $M$ is said to be nonnil-P-injective if, for any $a \in R\setminus \operatorname{Nil}(R)$, every homomorphism from $Ra$ to $M$ extends to a homomorphism from $R$ to $M$. Then $R$ is said to be a nonnil-P-coherent ring (resp., $\phi$-PF, nonnil-PP-ring) if, for any $a \in R \setminus \operatorname{Nil}(R)$, $Ra$ is a finitely presented (resp., flat, projective) module. In this paper, we study nonnil-P-coherent rings, $\phi$-PF-rings, and nonnil-PP-rings using $\phi$-P-flat and nonnil-P-injective modules.

Keywords: $phi$-P-flat, nonnil-P-injective, $phi$-PF-ring, nonnil-PP-ring, nonnil-P-coherent ring, P-coherent ring

MSC numbers: 13A15, 13C05, 13D30

Supported by: The authors would like to express their gratitude to the reviewer for the helpful suggestions. 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).