… has a phantom cover and an Ext-phantom preenvelope in R-… of phantom preenvelopes and
Ext-phantom precovers in R-Mod… bijective correspondence between an ℐ-preenvelope (resp. …

… of precovers and preenvelopes (… an Ext-phantom preenvelope (which is a monomorphism)
with the cokernel pure-projective. We note that the existence of Ext-phantom preenvelopes is …

… is an Ext-phantom morphism. We also investigate the existence of precovers and preenvelopes
by n-phantom and n-Ext-phantom morphisms. For instance, we prove that every left R-…

… Just as a phantom precover is called special [22, p. 230] … RD-phantom precover f : M → N
special if the kernel of f is RD-injective. Similarly, we will call an RD-Ext-phantom preenvelope : …

… We know that the concepts of preenvelopes and precovers (approximations) were introduced …
We note that the existence of Ext-phantom preenvelopes is a special case of the works [6, …

… of morphisms are used to define an extension |$\mathcal {I} \… [28] of a phantom preenvelope
in the stable homotopy … D] proved the existence of phantom precovers by a dual argument, …

… a cophantom morphism or Ext-phantom morphism [15, 22] if the induced morphism Ext …
preenvelope with a clean-projective cokernel. Finally, we establish the properties of precovers …

… ; A morphism g is an n-Ext-phantom morphism if and only if the … has an object-special n-phantom
precover. In addition, we … Dually we have the definitions of a c-preenvelope and a c…

… gate the interplay among special precovering ideals, special … any E-extension δ, we have
that ϕ⋆δ is an I⋆-extension. Let … that a is a special I⊥E -preenvelope of A. Since x ∈ I ⊆ ⊥E (I…

… phantom morphisms such as d-phantoms, d-Ext-phantoms, … Dually, we have the definitions
of a C-preenvelope and a C-… An I-precover is called I-cover if every endomorphism h of M …