ALGEBRA COLLOQUIUM, vol.29, no.02, pp.231-240, 2022 (SCI-Expanded)
Let A be an abelian category and M is an element of A. Then M is called a (D-4)-object if, whenever A and B are subobjects of M with M = A circle plus B and f : A -> B is an epimorphism, Ker f is a direct summand of A. In this paper we give several equivalent conditions of (D-4)-objects in an abelian category. Among other results, we prove that any object M in an abelian category A is (D-4) if and only if for every subobject K of M such that K is the intersection K-1 boolean AND K-2 of perspective direct summands K-1 and K-2 of M with M = K-1 + K-2, every morphism phi: M -> M/K can be lifted to an endomorphism theta: M -> M in End(A)(M).