We introduce and study relative Rickart objects and dual relative Rickart objects in abelian categories. We show how our theory may be employed in order to study relative regular objects and (dual) relative Baer objects in abelian categories. We also give applications to module and comodule categories.

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In this paper, we study the behavior of endomorphism rings of a cyclic, finitely presented module of projective dimension ≤ 1. This class of modules extends to arbitrary rings the class of couniformly presented modules over local rings.