This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

* General Introduction * Tools from Logic * Differential and Integral Calculus * General Topology * Functions of a Real Variable * Functions of a Complex Variable * Linear Spaces * Topological Groups and Lie Groups * Selected Topics * Concerning the History of the Calculus

Non-standard analysis

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

A branch of ordered algebraic structures has grown, motivated by $K$-theoretic applications and mainly concerned with partially ordered abelian groups satisfying the Riesz interpolation property This monograph is the first source in which the algebraic and analytic aspects of these interpolation groups have been integrated into a coherent framework for general reference The author provides a solid foundation in the structure theory of interpolation groups and dimension groups (directed unperforated interpolation groups), with applications to ordered $K$-theory particularly in mind Although interpolation groups are defined as purely algebraic structures, their development has been strongly influenced by functional analysis This cross-cultural development has left interpolation groups somewhat estranged from both the algebraists, who may feel intimidated by compact convex sets, and the functional analysts, who may feel handicapped by the lack of scalars This book, requiring only standard first-year graduate courses in algebra and functional analysis, aims to make the subject accessible to readers from both disciplines High points of the development include the following: characterization of dimension groups as direct limits of finite products of copies of the integers; the double-dual representation of an interpolation group with order-unit via affine continuous real-valued functions on its state space; the structure of dimension groups complete with respect to the order-unit norm, as well as monotone sigma-complete dimension groups and dimension groups with countably infinite interpolation; and an introduction to the problem of classifying extensions of one dimension group by another The book also includes a development of portions of the theory of compact convex sets and Choquet simplices, and an expository discussion of various applications of interpolation group theory to rings and $C^*$-algebras via ordered $K_0$ A discussion of some open problems in interpolation groups and dimension groups concludes the book Of interest, of course, to researchers in ordered algebraic structures, the book will also be a valuable source for researchers seeking a background in interpolation groups and dimension groups for applications to such subjects as rings, operator algebras, topological Markov chains, positive polynomials, compact group actions, or other areas where ordered Grothendieck groups might be useful This is a reprint of the 1986 original (SURV/20S)

Partially ordered abelian groups with interpolation

Commutative rings I

Introduction. Of the various spaces of ideal of rings that have been studied (see [1 ], for example) we are focusing attention on the space of minimal prime ideals because of its special role in the case of rings of continuous functions. For the simplest manifestation of this role, consider the compact space N* obtained from the discrete space of positive integers N by adjoining a single point, o. In the ring C(N*) of all continuous real-valued functions on N*, the maximal ideals are very easy to describe. They are in one-one correspondence with the points of N*, and are given by

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The Space of Minimal Prime Ideals of a Commutative Ring

An abstract ring in which all finitely generated ideals are principal will be called an F-ring. Let C(X) denote the ring of all continuous real-valued functions defined on a completely regular (Hausdorff) space X. This paper is devoted to an investigation of those spaces X for which C(X) is an F-ring. In any such study, one of the problems that arises naturally is to determine the algebraic properties and implications that result from the fact that the given ring is a ring of functions. Investigation of this problem leads directly to two others: to determine how specified algebraic conditions on the ring are reflected in topological properties of the space, and, conversely, how specified topological conditions on the space are reflected in algebraic properties of the ring. Our study is motivated in part by some purely algebraic questions concerning an arbitrary F-ring S-in particular, by some problems involving matrices over S. Continual application will be made of the results obtained in the preceding paper [4]. This paper will be referred to throughout the sequel as GH. We wish to thank the referee for the extreme care with which he read both this and the preceding paper, and for making a number of valuable suggestions. The outline of our present paper is as follows. In ?1, we collect some preliminary definitions and results. ?2 inaugurates the study of F-rings and F-spaces (i.e., those spaces X for which C(X) is an F-ring). The space of reals is not an F-space; in fact, a metric space is an F-space if and only if it is discrete. On the other hand, if X is any locally compact, a-compact space (e.g., the reals), then fX-X is an F-space. Examples of necessary and sufficient conditions for an arbitrary completely regular space to be an F-space are: (i) for every f C(X), there exists k E C(X) such that f k Jf J; (ii) for every maximal ideal M of C(X), the intersection of all the prime ideals of C(X) contained in M is a prime ideal. In ??3 and 4, we study Hermite rings and elementary divisor rings(2).

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Rings of continuous functions in which every finitely generated ideal is principal

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On the structure of a class of archimedean lattice-ordered algebras

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Some properties of compactifications

The present paper deals with two distinct, though related, questions, concerning the ring C(X, R) of all continuous real-valued functions on a completely regular topological space X. The first of these, treated in ? ?1-7, is the study of what we call P-spacesthose spaces X such that every prime ideal of the ring C(X, R) is a maximal ideal. The background and motivation for this problem are set forth in ?1. The results consist of a number of theorems concerning prime ideals of the ring C(X, R) in general, as well as a series of characterizations of P-spaces in particular. The second problem, discussed in ??8-10, is an investigation of what Hewitt has termed Q-spaces-those spaces X that cannot be imbedded as a dense subset of any larger space over which every function in C(X, R) can be continuously extended. An introduction to this question is furnished in ?8. Our discussion of Q-spaces is confined to the class of linearly ordered spaces (introduced in ?6). We are able to settle the question as to when an arbitrary linearly ordered space is or is not a Q-space. The concept of a paracompact space turns out to be intimately related to these considerations. We also derive a characterization of linearly ordered paracompact spaces, and we find in particular that every linearly ordered Q-space is paracompact. A result obtained along the way is that every linearly ordered space is countably paracompact.

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Melvin Henriksen

Papers

The Space of Minimal Prime Ideals of a Commutative Ring

Rings of continuous functions in which every finitely generated ideal is principal

On the structure of a class of archimedean lattice-ordered algebras

Some properties of compactifications

Concerning rings of continuous functions