# Studies in radical theory for restricted classes of rings

McConnell, N R(Nicholas Richard) (1990) Studies in radical theory for restricted classes of rings. PhD thesis, University of Tasmania.

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## Abstract

This thesis is a study of radical ideals in restricted domains of
associative rings.
The first chapter introduces a generalisation of the concept of strictness,
and studies, for a given radical class, the class of rings for which the radical is
hereditary and/or strict (in the general sense). This leads to some results on
embedding radical classes in generalised strict radical classes, and on degrees of
In the second chapter, the ideals of a ring which can be radical ideals are
studied, with characterisations being given for Dedekind domains and some of
their extension rings. This characterises radical ideals for a fairly wide class of
integral domains, and thus goes some way toward characterising supernilpotent
In the third chapter, a study is made of radical classes of commutative
rings, with a characterisation being given of the strongly hereditary strict
radicals, and the implications of restricting to commutativity are considered for
some other results and problems. There is also an investigation of the permanent
radical — that is, the part of the radical of a ring which remains as part of the
radical of any extension — for more general classes of rings.
The thesis thus gives an insight into the behaviour of radical classes