$RD$-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.