Let A be an abelian category and M∈A.Then M is called a(D_(4))-object if,whenever A and B are subobjects of M with M=A⊕B and f:A→B is an epimorphism,Ker f is a direct summand of A.In this paper we give several equi...Let A be an abelian category and M∈A.Then M is called a(D_(4))-object if,whenever A and B are subobjects of M with M=A⊕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)∩K_(2)of perspective direct summands K_(1)and K_(2)of M with M=K_(1)+K_(2),every morphismφ:M→M/K can be lifted to an endomorphismθ:M→M in End _(A)(M).展开更多
文摘Let A be an abelian category and M∈A.Then M is called a(D_(4))-object if,whenever A and B are subobjects of M with M=A⊕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)∩K_(2)of perspective direct summands K_(1)and K_(2)of M with M=K_(1)+K_(2),every morphismφ:M→M/K can be lifted to an endomorphismθ:M→M in End _(A)(M).
