A morphism f of left R-modules is called an RD-phantom morphism if the induced morphism Tor1(R=aR,f)=0 for any a  R. Similarly, a morphism g of left R-modules is said to be an RD-Ext-phantom morphism if the induced morphism Ext1(R=Ra,g)=0 for any a  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 Prüfer 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.