Abelian groups as artinian or noetherian modules above endomorphism rings

Abstract

The A and B Abellian groups, such that the Hom(A, B) homomorphism group is the Artin module over the ring of the B group endomorphism, are described. Description of the A and B group for which the Hom(A,B) group is the Artin module over the ring of the A group endomorphism is reduced to the case when the A group has no torsion and the B group is either a quasi-cyclic group or a divisible group without torsion. The A and B Abellian groups for which the Hom(A,B) group is the Neter module over the E(A) or E(B) ring are characterized. The research of arbitrary Abellian group with the link Neter ring of endomorphisms is reduced to the research of the group without torsion with the link Neter ring of endomorphisms. The research of the right Neter ring of endomorphisms remained uncompleted. The separable Abele groups without torsion with the link and right Neter rings of endomorphisms are described.