(D4)-Objects in Abelian Categories

Let 𝒜 be an abelian category and [Formula: see text]. Then M is called a [Formula: see text]-object if, whenever A and B are subobjects of M with [Formula: see text] and [Formula: see text] is an epimorphism, [Formula: see text] is a direct summand of A. In this paper we give several equivalent conditions of [Formula: see text]-objects in an abelian category. Among other results, we prove that any object M in an abelian category 𝒜 is [Formula: see text] if and only if for every subobject K of M such that K is the intersection [Formula: see text] of perspective direct summands [Formula: see text] and [Formula: see text] of M with [Formula: see text], every morphismr [Formula: see text] can be lifted to an endomorphism [Formula: see text] in [Formula: see text].