J. Korean Math. Soc. 2018; 55(5): 1221-1233
Online first article March 27, 2018 Printed September 1, 2018
https://doi.org/10.4134/JKMS.j170667
Copyright © The Korean Mathematical Society.
Wei Zhao
ABa Teachers university
Let $R$ be a commutative ring with non-zero identity and let $NN(R)=\{I\,|\,I$ is a nonnil ideal of $R\}$. Let $M$ be an $R$-module and let $\phi$-$\rm{tor}(M)=\{x\in M\,|\,Ix=0$ for some $I\in NN(R)\}$. If $\phi$-$\rm{tor}(M)=M$, then $M$ is called a $\phi$-torsion module. An $R$-module $M$ is said to be $\phi$-flat, if $0\rightarrow A\otimes_RM\rightarrow B\otimes_RM\rightarrow C\otimes_RM\rightarrow 0$ is an exact $R$-sequence, for any exact sequence of $R$-modules $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$, where $C$ is $\phi$-torsion. In this paper, the concepts of NRD-submodules and NP-submodules are introduced, and the $\phi$-flat modules over a $\phi$-Prufer ring are investigated.
Keywords: $\phi$-torsion module, $\phi$-flat module, $\phi$-Prufer ring
MSC numbers: Primary 13C05, 13C11, 13C12
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