Extensions of the Krull-Schmidt-Azumaya Theorem

Abstract

<p>The question which promoted this dissertation is the following: "How unique is a direct sum decomposition of a given R-module?" The classical result is tho direction is the so-called Krull-Schmitd-Azumaya Theorem, proved by Gorô Azumaya in [1]. It gives an answer to the question in the case that the given R-module is a direct sum of the sub-modules with local endomorphism ring. It is generalizations and extensions of this theorem that this paper is concerned with. The results of this thesis are stated and proved in a more general categorical setting than mod-R. Moreover, we do not resort to the embedding theorem, with the idea in mind that further generalization in those categories we are considering and similar results in other sort of categories may be suggested.</p> <p>Chapter I lays some necessary categorical ground-work. In Chapter 2 we combine results of S. B. Conlon [2] and S. Elliger [4] within our categorical setting to obtain a generalization of the Krull-Schmidt-Azumaya Theorem. We consider representations of an object as an essential extension of a direct sum of summands (rather than simply direct sum decompositions), and we allow certain summands other than those with local endomorphism ring. Chapter 3, follow [10], (which was in turn applying the results of [3]) extends the concept of "local endomorphism ring" to the concept of "the exchange property" and produces certain coproduct uniqueness theorems. Finally, in Chapter 4, we consider decomposition of injectives and we see that certain problems involving coproduct decompositions can be eliminated in the case where the objects concerned are injective. We present a uniqueness theorem due to R.B. Warfield [10] and draw conclusions from this with the aid of the "spectral category" (which will be define and examined).