Subgroups generated by images of endomorphisms of Abelian groups and duality

A subgroup H of a group G is called endo-generated if it is generated by endo-images, i.e. images of endomorphisms of G. In this paper we determine the following classes of Abelian groups: (a) the endo-groups, i.e. the groups all of whose subgroups are endo-generated; (b) the endo-image simple groups, i.e. the groups such that no proper subgroup is an endo-image; (c) the pure-image simple, i.e. the groups such that no proper pure subgroup is an endo-image; (d) the groups all of whose endo-images are pure subgroups; (e) the ker-gen groups, i.e. the groups all of whose kernels are endo-generated. Some dual notions are also determined.

Abelian groups whose nonzero endomorphisms have nonessential kernels

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.

Scott heights of abelian groups

The Scott height of a structure gives ordinal measure of the inhomogeneity of the structure. The Scott specturm of a collection of structures is the set of Scott heights of structures in the collection. We will recall the precise definitions of these and related concepts in the next section. The reader thoroughly unfamiliar with these notions may want to skip ahead before reading the rest of this Introduction.In [11] it is shown that every model of the complete theory of (, +, 1), where, as usual, denotes the integers, is ℵ0-homogeneous, and therefore has Scott height at most ω. On the other hand, a footnote in [1] gives a model of the theory of (, +) which is not ℵ0-homogeneous, while in [11] such a model is described which can be expanded to a model of the theory of (, +, 1). However, since it is also true that any model of the theory of (, +) is isomorphic to a subgroup, in fact a pure subgroup, of a direct sum of and a torsion-free divisible group, it is easy to see that any such model must be ≡∞ω to a model of cardinality at most and so must have Scott height below .After having recalled the relevant material about Scott heights in §2, we will review the situation for torsion abelian groups in §3. In §4 we shall produce torsion-free abelian groups of high Scott height. It is the proof of Theorem 15 that was our primary motivation in writing this paper.

Pure subgroups of LCA groups

This paper originated with our interest in the open question “If every pure subgroup of an LCA group G is closed, must G be discrete ?” that was raised by Armacost. The answer was surprisingly easy, but led to some interesting questions. We attempted to characterise those LCA groups that contain a proper pure dense subgroup, and found that every non-discrete torsion-free LCA group contains a proper pure dense subgroup; so does every non-discrete infinite self-dual torsion LCA group. We also give a necessary and sufficient condition for a torsion LCA group to contain a proper pure dense subgroup.

NEAT AND NEAT-HIGH EXTENSIONS

In the present paper, we find conditions that change a neat­ high subgroup into a pure-high subgroup and a neat subgroup, to a pure subgroup. It is shown that a neat-high extension can be transformed into a pure-high extension and a neat extension, to a pure extension. Furthermore, splitting conditions for a neat exact sequence are obtained.

Annihilator equivalence of torsion-free abelian groups

We define an equivalence relation on the class of torsion-free abelian groups under which two groups are equivalent ifevery pure subgroup of one has a non-zero image in the other, and each has a non-zero image in every torsion-free factor of the other.We study the closure properties of the equivalence classes, and the structural properties of the class of all equivalence classes. Finally we identify a class of groups which satisfy Krull-Schmidt and Jordan-Hölder properties with respect to the equivalence.

Quasi-P-Pure-Injective Groups

Recently, a great deal of attention has been paid to the concept of quasipure injectivity introduced by L. Fuchs as Problem 17 in [5]. An abelian group G is said to be quasi-pure-injective (q.p.i.) if every homomorphism from a pure subgroup of G to G can be lifted to an endomorphism of G. D. M. Arnold, B. O'Brien and J. D. Reid have succeeded in [1] to characterize torsion free q.p.i. of finite rank, whereas in [2] we solved the torsion case and in [3] we studied certain classes of infinite rank torsion free q.p.i. groups.

Pure N-high Subgroups, P-adic Topology and Direct Sums of Cyclic Groups

This paper is divided into two sections. In the first, we characterize the subgroups N of a reduced abelian primary group for which all pure N-high subgroups are bounded. This condition on pure N-high subgroups occurs in several instances, for instance, all pure N-high subgroups of a primary group G are bounded if G is the smallest pure subgroup of G containing N; all N-high subgroups are bounded if N ≠ 0 and all N-high subgroups are closed in the p-adic topology.