Foundations of Commutative Rings and Their Modules - Wang, Fanggui;Kim, Hwankoo
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This book provides an introduction to the foundations and recent developments in commutative algebra. A look at the contents of the first five chapters shows that the topics covered are those usually found in any textbook on commutative algebra. However, this book differs significantly from most commutative algebra textbooks: namely in its treatment of the Dedekind-Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings, the valuative dimension, and the Nagata rings. Chapter 6 goes on to present w-modules over commutative rings, as they are most…mehr

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Produktbeschreibung
This book provides an introduction to the foundations and recent developments in commutative algebra. A look at the contents of the first five chapters shows that the topics covered are those usually found in any textbook on commutative algebra. However, this book differs significantly from most commutative algebra textbooks: namely in its treatment of the Dedekind-Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings, the valuative dimension, and the Nagata rings. Chapter 6 goes on to present w-modules over commutative rings, as they are most commonly used in torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of pullbacks, especially Milnor squares and D + M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings of finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated by combining homological tricks and methods of star operation theory. Chapter 10 is devoted to the Grothendieck group of a commutative ring. In particular, the Bass-Quillen problem is discussed. Finally, Chapter 11 introduces relative homological algebra, especially where the related notions of integral domains appearing in classical ideal theory are defined and studied using the class of Gorenstein projective modules. In Chapter 12, in this new edition, properties of cotorsion theories are introduced and show, for any cotorsion pair, how to construct their homology theory. Each section of the book is followed by a selection of exercises of varying difficulty. This book appeals to a wide readership, from graduate students to academic researchers interested in studying commutative algebra.
Autorenporträt
Hwankoo Kim is a professor at School of Computer Engineering of Hoseo University. Fanggui Wang is a professor at College of Mathematics of Sichuan Normal University.
Rezensionen
"The authors present the bulk of the traditional material concerning the theory of commutative rings and their modules, incorporating recent discoveries on this subject. ... The reviewer recommends this volume to graduate students and to academic researchers who are interested in studying commutative algebra from a not necessarily Noetherian point of view." (Marco Fontana, Mathematical Reviews, November, 2017)

"The book under review is intended to serve as a rather comprehensive textbook for a course in commutative algebra at the graduate level as well as a reference book for researchers. ... The main advantage of the book under review in compare to similar books is the comprehension and detailed arguments that would be of great benefit to the beginners or experts in other fields. ... could well be listed among recommended literature for students and researchers from other fields." (Siamak Yassemi, zbMATH 1367.13001, 2017)