(D 4)-Objects in Abelian Categories

Let be an abelian category and M A. Then M is called a (D4)-object if, whenever A and B are subobjects of M with M = A B and f:A→B is an epimorphism, Kera f is a direct summand of A. In this paper we give several equivalent conditions of (D4)-objects in an abelian category. Among other results, we prove that any object M in an abelian category is (D4) if and only if for every subobject K of M such that K is the intersection K1K2 of perspective direct summands K1 and K2 of M with M = K1 + K2, every morphismr φ: M → M/K can be lifted to an endomorphism θ:M→M in EndA(M).