Foundations of Commutative Rings and Their Modules

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 DedekindMertens 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 BassQuillen 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.

Adds the w-theory point of view in Chaps. 612 while maintaining the characteristics of the individual chapters Deals with the classical ring and module theory but contains new results not found in other textbooks Is the only book to include both Gorenstein homological algebra and the cotorsion theory

Autorentext

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.

Inhalt

Basic Theory of Rings and Modules.- Several Classical Module Classes in the Module Category.- Homological Methods.- Basic Theory of Noetherian Rings.- Extensions of Rings.- w-Modules over Rings.- Multiplicative Ideal Theory over Integral Domains.- Structural Theory of Milnor Squares.- Coherent Rings with Finite Weak Global Dimension.- Grothendieck Groups of Rings.- Relative Homological Algebra.- Cotorsion Theory.