The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

Tariel Kemoklidze

doi:10.4236/apm.2013.38090

Abstract

The paper considers the lattice of fully invariant subgroups of the cotorsion hull when a separable primary group T is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation difined on the set of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.

Highlights

We consider questions of the theory of abelian groups and throughout the paper the word “group” means an additively written abelian group
Of the group A is called fully invariant if for any endomorphism of the group A this subgroup B is mapped into B
Though the notion of a cotorsion group and its generalizations are studied sufficiently well, little is known about the lattice of fully invariant subgroups of a cotorsion group

Summary

Introduction

We consider questions of the theory of abelian groups and throughout the paper the word “group” means an additively written abelian group. Though the notion of a cotorsion group and its generalizations are studied sufficiently well (see [15,16,17,18]), little is known about the lattice of fully invariant subgroups of a cotorsion group. The lattice of fully invariant subgroups is studied by means of indicators (see [2, Theorem 67.1]). A. Mader [11] showed that an algebraically compact group is fully transitive and described by means of indicators of the lattice of fully invariant subgroups of an algebraically compact group. Moskalenko [13] proved that when T is the direct sum of cyclic p -groups, T is fully transitive and all the conditions of Theorem 1.1 are fulfilled In this case, too, the lattice * of indicator filters describes the lattice of fully invariant subgroups. T is an arbitrary direct sum of torsioncomplete groups and the lower semilattice is defined by a simpler new relation (see Definition 2.2) which makes it easier to verify the properties of Theorem 1.1

A Semilattice

The Lattice of Fully Invariant Subgroups of the Group T

H Bi ai0