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 …

A morphism f: M→ N of left R-modules is called a phantom morphism if the induced
morphism for every (finitely presented) right R-module A. Similarly, a morphism g: M→ N of …

A morphism f: M→ N of left R-modules is called an n-phantom morphism if the induced
morphism Tor n R (A, f)= 0 for every (finitely presented) right R-module A. Analogously, a …

A morphism f: M→ N f:M→N of left R-modules is called an n-phantom morphism (resp. a Tor
n-epimorphism) if the induced morphism Tor n (A, f)= 0 (resp. Tor n (A, f) is an epimorphism) …

Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be (m,
n)-injective if Ext1 (P, M)= 0 for any (m, n)-presented right R-module P; M is said to be (m, n) …

Let ℱ 𝒫∞ R be the class of all left R-modules M which has a projective resolution by finitely
generated projectives. An exact sequence 0→ A→ B→ C→ 0 of right R-modules is called …

Let $ R $ be a ring. A morphism $\alpha: X\rightarrow Y $ of left $ R $-modules is called a
finitely phantom morphism if for each morphism ${\beta: F\rightarrow X} $ with $ F $ finitely …

Let R be a commutative ring with a nonzero identity and M an R-module. Set ϕ-tor (M)={x∈
M| sx= 0 for some s∈ R∖ Nil (R)}, if ϕ-tor (M)= M then M is called a ϕ-torsion module. An R …

This paper extends many conclusions based on phantom envelopes and Ext-phantom
covers of modules, and we find that many important properties still hold after replacing …

Abstract A right R-module M is called kernel-invariant if the kernels of all endomorphisms of
M are invariant under all automorphisms of M, and R is called rightkernel-invariant if RR is …