$(n,d)$-$\mathcal{X}_R$-phantom and $(n,d)$-$_R\mathcal{X}$-cophantom morphisms

Several authors have been interested in some like phantom morphisms such as $d$-phantoms, $d$-$\operatorname{Ext}$-phantoms, neat-phantom morphisms, clean- cophantom morphisms, $RD$-phantom morphisms and $RD$-$\operatorname{Ext}$-phantom morphisms. In this paper, we prove that these notions can be unified. We are mainly interested in proving that the majority of the existing results hold true in our general framework.

Keywords:

$(n, d)$-$\mathcal{X}_R$-phantom morphism, $(n d)$-$_R\mathcal{X}$-cophantom morphism, $n$-$\mathcal{X}$-coherent ring,

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