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Rd-phantom and RD-Ext-phantom morphisms
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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 Pr?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.
Title: Rd-phantom and RD-Ext-phantom morphisms
Description:
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 Pr?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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