On $\Phi$-flat modules and $\Phi$-Pr\"{u}fer rings
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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.
Abstract
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.
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