F -Copartial Morphisms

In this paper, first we give the definition of F-copartial morphisms with an additive exact substructure F of an exact structure E in an additive category A. Then, we study many properties of F-copartial morphisms. Moreover, we define F-copartial morphisms with a pure-exact structure F and with a finite pure-exact structure F in the category of modules over a ring and call them copartial morphisms and finitely copartial morphisms, respectively. We also investigate the relations between them and give the new characterizations of finitely (singly) pure-projective modules, flat modules and finitely (singly) projective modules with copartial morphisms and finitely copartial morphisms. Finally, we define μ-partial morphisms for a defining matrix μ and give a new characterization of semi-compact modules with μ-partial morphisms.