### $RD$-phantom and $RD$-$\Ext$-phantom morphisms

#### Abstract

A morphism $f$ of left $R$-modules is called an

$RD$-phantom morphism if the induced morphism $\Tor_{1}(R/aR,f)=0$

for any $a\in R$. Analogously, a morphism $g$ of left $R$-modules is

said to be an $RD$-$\Ext$-phantom morphism if the induced morphism

$\Ext^{1}(R/Ra,g)=0$ for any $a\in R$. It is proven that a morphism

$f$ is an $RD$-phantom morphism if and only if the pullback of any

short exact sequence along $f$ is an $RD$-exact sequence; a morphism

$g$ is an $RD$-$\Ext$-phantom morphism if and only if the pushout of

any short exact sequence along $g$ is an $RD$-exact sequence. We

also characterize Pr\"{u}fer domains, left $P$-coherent rings, left

$PP$ rings, von Neumann regular rings in terms of $RD$-phantom and

$RD$-$\Ext$-phantom morphisms. Finally we prove that every module

has an epic $RD$-phantom cover with the kernel

$RD$-injective and has a monic $RD$-$\Ext$-phantom preenvelope with the cokernel $RD$-projective.

$RD$-phantom morphism if the induced morphism $\Tor_{1}(R/aR,f)=0$

for any $a\in R$. Analogously, a morphism $g$ of left $R$-modules is

said to be an $RD$-$\Ext$-phantom morphism if the induced morphism

$\Ext^{1}(R/Ra,g)=0$ for any $a\in R$. It is proven that a morphism

$f$ is an $RD$-phantom morphism if and only if the pullback of any

short exact sequence along $f$ is an $RD$-exact sequence; a morphism

$g$ is an $RD$-$\Ext$-phantom morphism if and only if the pushout of

any short exact sequence along $g$ is an $RD$-exact sequence. We

also characterize Pr\"{u}fer domains, left $P$-coherent rings, left

$PP$ rings, von Neumann regular rings in terms of $RD$-phantom and

$RD$-$\Ext$-phantom morphisms. Finally we prove that every module

has an epic $RD$-phantom cover with the kernel

$RD$-injective and has a monic $RD$-$\Ext$-phantom preenvelope with the cokernel $RD$-projective.

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