Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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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.

On $\phi$-exact sequences and $\phi$-projective modules

Wei Zhao

ABa Teachers University

Abstract

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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