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UBC Theses and Dissertations

Rings with a polynomial identity Bridger, Lawrence Ernest

Abstract

Since Kaplansky's first paper on the subject of P.I. rings appeared in 1948, many fruitful results have arisen from the study of such rings. This thesis attempts to present the most important of these results in a unified theory. Chapter I gives the basic notation, definitions, a number of small lemmas together with Kaplansky's incisive result on primitive P.I. rings. We investigate also the Kurosh problem for P.I. rings, providing for such rings an affirmative answer. A rather nice universal property for P.I. rings which ensures that all P.I. rings satisfy some power of the standard identity is proved. Chapter II deals with particular types of rings such as rings without zero divisors and prime rings and culminates in a pair of pretty results due to Posner and Procesi. We show that prime P.I. rings have a rather tight structure theory and in fact the restrictions on the underlying set of coefficients can in this case be relaxed to a very great extent. Chapter III is exclusively devoted to P.I. rings with involution. Although such rings are rather specialized much has been accomplished in this direction in recent years and many beautiful theorems and proofs have been established, especially by Amitsur and Martindale. The source material for chapter IV is primarily Procesi and Amitsur's work on Jacobson rings and Hilbert algebras. Application to Hilbert's Nullstellensatz and to the Burnside problem are considered. Finally, chapter V concerns itself completely with generalizations of the preceding four chapters. For the most part these results do not generalize entirely, but by reducing our demand on polynomial identities slightly, many remarkably fine results have been proved.

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