J. Korean Math. Soc. 2021; 58(6): 1513-1528
Online first article September 7, 2021 Printed November 1, 2021
https://doi.org/10.4134/JKMS.j210180
Copyright © The Korean Mathematical Society.
Wei Zhao
ABa Teachers University
Let $R$ be a commutative ring with prime nilradical $Nil(R)$ and $M$ an $R$-module. Define the map $\phi: R\rightarrow R_{Nil(R)}$ by $\phi (r) = \frac{r}{1}$ for $r\in R$ and $\psi: M\rightarrow M_{Nil(R)}$ by $\psi (x) = \frac{x}{1}$ for $x\in M$. Then $\psi(M)$ is a $\phi(R)$-module. An $R$-module $P$ is said to be $\phi$-projective if $\psi(P)$ is projective as a $\phi(R)$-module. In this paper, $\phi$-exact sequences and $\phi$-projective $R$-modules are introduced and the rings over which all $R$-modules are $\phi$-projective are investigated.
Keywords: $\phi$-exact sequence, nonnil-divisible module, $\phi$-projective module
MSC numbers: Primary 13C05, 13C10, 13C12
Supported by: This work was financially supported by the National Natural Science Foundation of China 12061001, 11861001, the China Postdoctoral Science Foundation 2021M691526, the Science and Technology Plan Project of Aba Prefecture 20RKX0001, and Aba Teachers University ASB20-02, ASA20-02, ASC20-02, 201901011, 201907019, 201910107, 201910108.
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