THE THEORY OF RINGS
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The theory that forms the subject of this book had its beginning with Artin's extension in 1927 of Wedderburn's structure theory of algebras to rings satisfying the chain conditions. Since then the theory has been considerably extended and simplified. The only exposition of the subject in book form that has appeared to date is Deuring's Algebren published in the Ergebnisse series in 1935. Much progress has been made since then and this perhaps justifies a new exposition of the subject.
The present account is almost completely selfcontained. That this has been possible in a book dealing with results of the significance of Wedderburn's theorems, the AlbertBrauerNoether [A Adrain Albert, Richard Brauer, Emmy Noether] theory of simple algebras and the arithmetic ideal theory is another demonstration of one of the most remarkable characteristics of modern algebra, namely, the simplicity of its logical structure.
Roughly speaking our subject falls into three parts: structure theory, representation theory and arithmetic ideal theory. The first of these is an outgrowth of the structure theory of algebras. It was motivated originally by the desire to discover and to classify "hypercomplex" extensions of the field of real numbers. The most important names connected with this phase of the development of the theory are those of Molien, Dedekind, Frobenius and Cartan. The structure theory for algebras over a general field dates from the publication of Wedderburn's thesis in 1907; the extension to rings, from Artin's paper in 1927. The theory of representations was originally concerned with the problem of representing a group by matrices. This was extended to rings and was formulated as a theory of modules by Emmy Noether. The study of modules also forms an important part of the arithmetic ideal theory. This part of the theory of rings had its origin in Dedekind's ideal theory of algebraic number fields and more immediately in Emmy Noether's axiomatic foundation of this theory.
Throughout this book we have placed particular emphasis on the study of rings of endomorphisms. By using the regular representations the theory of abstract rings is obtained as a special case of the more concrete theory of endomorphisms. Moreover, the theory of modules, and hence representation theory, may be regarded as the study of a set of rings of endomorphisms all of which are homomorphic images of a fixed ring R. Chapter 1 lays the foundations of the theory of endomorphisms of a group. The concepts and results developed here are fundamental in all the subsequent work. Chapter 2 deals with vector spaces and contains some material that, at any rate in the commutative case, might have been assumed as known. For the sake of completeness this has been included. Chapter 3 is concerned with the arithmetic of noncommutative principal ideal domains. Much of this chapter can be regarded as a special case of the general arithmetic ideal theory developed in Chapter 6. The methods of Chapter 3 are, however, of a much more elementary character and this fact may be of interest to the student of geometry, since the results of this chapter have many applications in that field. A reader who is primarily interested in structure theory or in representation theory may omit Chapter 3 with the exception of 3. Chapter 4 is devoted to the development of these theories and to some applications to the problem of representation of groups by projective transformations and to the Galois theory of division rings. In Chapter 5 we take up the study of algebras. In the first part of this chapter we consider the theory of simple algebras over a general field. The second part is concerned with the theory of characteristic and minimum polynomials of an algebra and the trace criterion for separability of an algebra.
In recent years there has been a considerable interest in the study of rings that do not satisfy the chain conditions but instead are restricted by topological or metric conditions. We mention von Neumann and Murray's investigation of rings of transformations in Hilbert space, von Neumann's theory of regular rings and Gelfand's theory of normed rings. There are many important applications of these theories to analysis. Because of the conditions that we have imposed on the rings considered in this work, our discussion is not directly applicable to these problems in topological algebra. It may be hoped, however, that the methods and results of the purely algebraic theory will point the way for further development of the topological algebraic theory.
This book was begun during the academic year 19401941 when I was a visiting lecturer at Johns Hopkins University. It served as a basis of a course given there and it gained materially from the careful reading and criticism of Dr Irving Cohen who at that time was one of the auditors of my lectures. My thanks are due to him and also to Professors A A Albert, Schilling and Hurewicz for their encouragement and for many helpful suggestions.
N JACOBSON
Chapel Hill, N. C., March 7, 1943.
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