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Some topics in the theory of ring structures on Abelian groups

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Jackett, DR (1977) Some topics in the theory of ring structures on Abelian groups. PhD thesis, University of Tasmania.

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Abstract

In recent years, Fuchs has described the absolute annihilator and
the absolute (Jacobson) radical of a torsion group, and Gardner has
characterised the absolute annihilator of a completely decomposable
torsion-free group. In this thesis the problem of describing the
absolute annihilator and the absolute radical of certain abelian groups
is considered. This will involve a discussion of the rings on these
groups, and the information so obtained allows us to answer several other
questions from the theory of ring structures on abelian groups.
Complete descriptions of the absolute annihilator are given for
vector groups, separable groups, certain mixed groups of torsion-free
rank one, reduced algebraically compact groups, cohesive groups, and
reduced groups whose quotients mod torsion subgroups are divisible.
Partial characterisations are also provided for cotorsion groups, and
torsion-free groups of rank two. For the absolute radical of a group,
complete descriptions are provided for certain mixed groups of torsion-free rank one, reduced algebraically compact groups, certain strongly indecomposable
torsion-free groups of finite rank, and partial descriptions are given for
completely decomposable torsion-free groups, cotorsion groups, torsion-free
groups of rank two, and cohesive groups.
The properties of rings on some of the forementioned torsion-free
groups lead us to consider various aspects of nilpotence. Of particular
interest are the T-nilpotent rings on completely decomposable torsion-free
groups. A bound is also provided for the nil-degree, if it is finite, of
certain torsion-free groups. The mixed groups of torsion-free rank one
discussed in this thesis motivate an investigation of the additive group
of a regular ring. A question of Fuchs concerning these groups is
answered in the negative.

Item Type: Thesis (PhD)
Keywords: Group rings, Abelian groups
Copyright Holders: The Author
Copyright Information:

Copyright 1977 the Author - The University is continuing to endeavour to trace the copyright owner(s) and in the meantime this item has been reproduced here in good faith. We would be pleased to hear from the copyright owner(s).

Additional Information:

Thesis (Ph.D.)--University of Tasmania, 1978. Bibliography: l. 140-145

Date Deposited: 19 Dec 2014 02:23
Last Modified: 04 Apr 2017 22:53
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