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Radical theory in categorical settings Melvin, Eileen

Abstract

The basic theme of the thesis is the development of a theory of radicals in a categorical setting. Guided by the general theory of radicals in rings as presented by Divinsky in his book Rings and Radicals, [1], we have tried to isolate the minimal categorical assumptions requisite for a "decent" theory of radicals. The primary focus is on the duality between radical and semi-simple classes of objects, and the extendability, or non-extendability, of various properties of the radicals of rings to our two types of categories. In more detail then, the initial chapter essentially lists those, and only those, basic categorical propositions that we use. Most of these are in Mitchell, [4], but a few do not seem to have been formulated in the standard texts. In Chapter One, motivated by a paper on radical subcategories by Mary Gray,[2], we have approached the question from the point of view of determining the minimal categorical assumptions underlying the definition of radical. Here we introduce the notion of an A-category, which is rather weaker than that of the motivating, concept - the "semi-abelian" categories of [2], We formulate the notion of a radical property in this setting and show that we can obtain the principal theorem of the Gray paper with these weaker assumptions. Chapter Two contains the most technical results and begins with the introduction of our basic concept - a B-category. This is motivated by Sulinski's paper on categorical Brown-McCoy radicals, [5]. Guided by the axioms listed there, we have isolated those which seem sufficient for our purposes and proved the most useful properties. We then show that most of the general radical properties given in Divinsky's book extend to B-categories. Further, we show that in a B-category the notion of radical can be reformulated in such a way that an attractive duality between radical classes and semi-simple classes arises. The question of duality is treated separately in Chapter Three. We show that the construction of upper radical classes is intimately bound to this duality. In the purely ring setting of [1] this duality is rather obscured. The fourth chapter concerns itself with slightly more refined radical properties, again in the setting of a B-category. We examine the notion of "hereditary" properties, and show how the notion of "heart" and its relation to radicals can be extended to our categories. While the thrust of the first four chapters has been theoretical, Chapter Five is quite explicit. Here we construct an example which shows that two of .the major properties of radicals of rings do not extend to B-categories. This example is drawn from the category of finite-dimensional Lie Algebras and is a twenty-one dimensional Lie algebra over the prime field of characteristic three. The motivation for this example comes from an example of Jacobson in "Lie Algebras", [3], With this example our treatment of the general properties of rings as found in Divinsky is complete, for we have been able to decide for essentially all the major properties whether or not they are extendable to B-categories.

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