Abelian groups whose nonzero endomorphisms have nonessential kernels

Abstract

In this paper, we describe completely the [Formula: see text]-singular subgroup of an abelian group and a [Formula: see text]-nonsingular abelian group in terms of the basic subgroups of its [Formula: see text]-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a [Formula: see text]-nonsingular abelian group are [Formula: see text]-nonsingular and give a condition under which a pure extension of a [Formula: see text]-nonsingular abelian group by a [Formula: see text]-nonsingular group is [Formula: see text]-nonsingular.