Endomorphisms of countable direct sums of torsion complete groups

Abstract

This paper is a further study of those abelian p-groups G whose ring of endomorphisms is generated by its group of automorphisms. It was shown in [1 ] that if G is a direct sum of countable p-groups or a torsion complete p-group, p # 2, then each endomorphism of G was a sum of two automorphisms of G. In a later paper HILL [2] extended the result for direct sums of countable p-groups to the class of totally projective p-groups. Whether this property holds for direct sums of torsion complete p-groups is not known. In this paper it is shown that a countable direct sum of torsion complete p-groups does have the property that each endomorphism is a sum of two automorphisms. Throughout, p denotes a fixed odd prime. The socle of G is 6[p] = {xlpx = 0}. If/~ is a torsion complete p-group and B is a basic subgroup of/~ we write