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Mediterranean Journal Of Mathematics, vol.18, no.5, pp.1-25, 2021 (Journal Indexed in SCI Expanded)
In this paper, (dual) purely Rickart objects are introduced as generalizations of (dual) Rickart objects in Grothendieck categories. Examples showing the relations between (dual) relative Rickart objects and (dual) relative purely Rickart objects are given. It is shown that in a spectral category (dual) relative purely Rickart objects coincide with (dual) relative Rickart objects. (Co)products of (dual) relative purely Rickart objects are studied. Classes all of whose objects are (dual) relative purely Rickart are identified. It is shown how this theory may be employed in order to study (dual) relative purely Baer objects in Grothendieck categories. Also applications to module and comodule categories are given.