The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

Tariel Kemoklidze

Department of Mathematics, Akaki Tsereteli State University, Kutaisi, Georgia.

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.

Keywords

Lattice of Fully Invariant Subgroups; Direct Sum of Torsion-Complete Groups; Cotorsion Hull

Conflicts of Interest

The authors declare no conflicts of interest.

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