On $\phi$-exact sequences and $\phi$-projective modules
J. Korean Math. Soc.
Published online September 7, 2021
wei zhao
ABa Teachers University
Abstract : Let $R$ be a commutative ring with prime nilradical $Nil(R)$ and $M$be 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 $\phi(R)$-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 : 13C05,13C10,13C12
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