Foundations of Commutative Rings and Their Modules

ISBN-13: 9789811098468 / Angielski / Miękka / 2018 / 699 str.

Fanggui Wang; Hwankoo Kim

Recenzje

"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) 

Spis treści

1 Basic Theory of Rings and Modules.- 2 The Category of Modules.- 3 Homological Methods.- 4 Basic Theory of Noetherian Rings.- 5 Extensions of Rings.- 6 w-Modules over Commutative Rings.- 7 Multiplicative Ideal Theory over Integral Domains.- 8 Structural Theory of Milnor Squares.- 9 Coherent Rings with Finite Weak Global Dimension.- 10 The Grothendieck Group of a Ring.- 11 Relative Homological Algebra.- References.- Index of Symbols.- Index.

Blurb

This book provides an introduction to the basics and recent developments of commutative algebra. A glance at the contents of the first five chapters shows that the topics covered are ones that usually are included in any commutative algebra text. However, the contents of this book differ significantly from most commutative algebra texts: namely, its treatment of the Dedekind–Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings and the valuative dimension, and Nagata rings. Going further, Chapter 6 presents w-modules over commutative rings as they can be most commonly used by torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of the pullbacks, especially Milnor squares and D+M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings with finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated on 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 aims to introduce relative homological algebra, especially where the related concepts of integral domains which appear in classical ideal theory are defined and investigated by using the class of Gorenstein projective modules. Each section of the book is followed by a selection of exercises of varying degrees of difficulty. This book will appeal to a wide readership from graduate students to academic researchers who are interested in studying commutative algebra.