UBC Theses and Dissertations

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UBC Theses and Dissertations

Some results in the theory of radicals of associative rings Heinicke, Allan George


Several aspects of the theory of radical classes in associative ring theory are investigated. In Chapter three, the Andrunakievic-Rjabuhin construction of radicals by means of annihilators of modules is employed to define several radical properties. One of these is shown to be the "weak radical" of Koh and Mewborn. The relations between these radicals, their properties and some of their applications to the study of classical quotient rings are investigated. In Chapter four, the ideals of a ring K of the form R(K), for a hereditary radical, R, are studied. A closure operation on the lattice of ideals is introduced, and the "closed" ideals are precisely the ideals of this type. It is proved that the ascending and descending chain conditions on the closed ideals of a ring imply that the ring has only a finite number of closed ideals. In Chapter five, finite subdirect sums of rings are studied. The properties of hereditary radicals and of the various structure spaces, in a situation where one has a finite subdirect sum of rings, are investigated.

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