Tsit-Yuen Lam Livres

This new book can be read independently from the first volume and may be used for lecturing, seminar- and self-study, or for general reference. It focuses more on specific topics in order to introduce readers to a wealth of basic and useful ideas without the hindrance of heavy machinery or undue abstractions. User-friendly with its abundance of examples illustrating the theory at virtually every step, the volume contains a large number of carefully chosen exercises to provide newcomers with practice, while offering a rich additional source of information to experts. A direct approach is used in order to present the material in an efficient and economic way, thereby introducing readers to a considerable amount of interesting ring theory without being dragged through endless preparatory material.

Focusing on basic ring theory, this textbook is designed for a one-semester course and is accessible to novices. It covers essential topics such as semisimple rings, Jacobson's radical, and representation theory, among others. The author emphasizes examples and motivation to enhance understanding, making it suitable for both graduate courses and self-study. The new edition includes over 400 exercises to reinforce comprehension of the material, ensuring a comprehensive learning experience for students interested in noncommutative rings.

“Serre’s Conjecture” refers to a statement made by J.-P. Serre in 1955 regarding whether finitely generated projective modules are free over a polynomial ring k[x1, ..., xn], where k is a field. This question arose from the analogy with affine schemes, where the affine n-space over k is contractible, leading to only trivial bundles. The inquiry was whether a similar result held in algebraic geometry. In this context, algebraic vector bundles over Spec k[x1, ..., xn] correspond to finitely generated projective modules over k[x1, ..., xn], making the question equivalent to determining if such projective modules are free for any base field k. Serre framed his statement as an open problem within the emerging sheaf-theoretic framework of algebraic geometry in the mid-1950s. He did not speculate on the outcome in his published work. However, a presumed positive answer to his question quickly became known as “Serre’s Conjecture.” Interest in this conjecture grew further with the development of two related fields: homological algebra and algebraic K-theory.

" This useful book, which grew out of the author's lectures at Berkeley, presents some 400 exercises of varying degrees of difficulty in classical ring theory, together with complete solutions, background information, historical commentary, bibliographic details, and indications of possible improvements or generalizations. The book should be especially helpful to graduate students as a model of the problem-solving process and an illustration of the applications of different theorems in ring theory. The author also discusses "the folklore of the the 'tricks of the trade' in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference". The problems are from the following the Wedderburn-Artin theory of semisimple rings, the Jacobson radical, representation theory of groups and algebras, (semi)prime rings, (semi)primitive rings, division rings, ordered rings, (semi)local rings, the theory of idempotents, and (semi)perfect rings. Problems in the areas of module theory, category theory, and rings of quotients are not included, since they will appear in a later book. "(T. W. Hungerford, Mathematical Reviews)