On 𝒮-phantom and simple-phantom morphisms

In this paper, we introduce and investigate two classes of morphisms relative to simple modules: [Formula: see text]-phantom morphisms, defined by the vanishing of [Formula: see text] for all simple right [Formula: see text]-modules [Formula: see text], and simple-phantom morphisms, characterized by the property that every composition [Formula: see text], with [Formula: see text] and [Formula: see text] simple, factors through a projective module. Unlike classical phantom morphisms, these two notions are shown to be genuinely distinct. By using the frameworks of [Formula: see text]-pure and neat exact sequences, we provide various characterizations of both classes and determine ring-theoretic conditions under which the two notions coincide.